remarkRemark hypothesisHypothesis claimClaim Reducing the boundary error in
numerical homogenizationSean P. Carney, Milica Dussinger, and Björn Engquist

# On the nature of the boundary resonance error in numerical homogenization
and its reduction ††thanks:  This research is supported by National Science
Foundation Grant DMS-1620396.

Sean P. Carney111Department of Mathematical Sciences, George Mason University,
Fairfax, VA, 22030, USA<EMAIL_ADDRESS>Milica
<EMAIL_ADDRESS>Björn Engquist333Department of
Mathematics and Oden Institute for Computational Engineering and Sciences, The
University of Texas at Austin, Austin, TX, 78712, USA;
<EMAIL_ADDRESS>
###### Abstract

Numerical homogenization of multiscale equations typically requires taking an
average of the solution to a microscale problem. Both the boundary conditions
and domain size of the microscale problem play an important role in the
accuracy of the homogenization procedure. In particular, imposing naive
boundary conditions leads to a $\mathcal{O}(\epsilon/\eta)$ error in the
computation, where $\epsilon$ is the characteristic size of the microscopic
fluctuations in the heterogeneous media, and $\eta$ is the size of the
microscopic domain. This so-called boundary, or “cell resonance” error can
dominate discretization error and pollute the entire homogenization scheme.
There exist several techniques in the literature to reduce the error. Most
strategies involve modifying the form of the microscale cell problem. Below we
present an alternative procedure based on the observation that the resonance
error itself is an oscillatory function of domain size $\eta$. After
rigorously characterizing the oscillatory behavior for one dimensional and
quasi-one dimensional microscale domains, we present a novel strategy to
reduce the resonance error. Rather than modifying the form of the cell
problem, the original problem is solved for a sequence of domain sizes, and
the results are averaged against kernels satisfying certain moment conditions
and regularity properties. Numerical examples in one and two dimensions
illustrate the utility of the approach.

###### keywords:

multiscale methods, numerical homogenization, boundary resonance error

35B27, 35B30, 65N06, 74Q05

##  1 Introduction

This work is concerned with developing methods for the numerical
homogenization of the multiscale elliptic equation

(1) $\displaystyle-\nabla\cdot\left(a^{\epsilon}\nabla u^{\epsilon}\right)$
$\displaystyle=f,\qquad\Omega\subset\mathbb{R}^{d}$ $\displaystyle
u^{\epsilon}$ $\displaystyle=g,\qquad\partial\Omega$

where $0<\epsilon\ll 1$, $|\Omega|=\mathcal{O}(1)$, and $a^{\epsilon}$ is
assumed to be symmetric, positive-definite, and bounded, so that $\forall
w\in\mathbb{R}^{d}$, $x\neq 0$,

(2) $\lambda|w|^{2}\leq\sup_{x\in\Omega}\left\langle
w,a^{\epsilon}(x)w\right\rangle\leq\Lambda|w|^{2},$

for some $0<\lambda<\Lambda<\infty$. $a^{\epsilon}$ is also assumed to
oscillate rapidly, that is, with frequencies $\sim 1/\epsilon$, which in turn
implies the solution to the elliptic equation $u^{\epsilon}$ is oscillatory as
well. Equation (1) models, for instance, steady-state heat conduction in a
composite material [45], as well as porous media flow through material with
highly variable permeability [21].

The mathematical study of the behavior of the solution to (1) as $\epsilon\to
0$ has a long history; see [15, 39, 19]. It is well known that when
$a^{\epsilon}$ is periodic, $u^{\epsilon}\to\overline{u}$ weakly in
$H^{1}(\Omega)$ and strongly in $L^{2}(\Omega)$, where $\overline{u}$
satisfies the homogenized equation

$\displaystyle-\nabla\cdot\left(\overline{a}\,\nabla\,\overline{u}\right)$
$\displaystyle=f,\qquad\Omega$ $\displaystyle\overline{u}$
$\displaystyle=g,\qquad\partial\Omega.$

The entries in the homogenized tensor $\overline{a}$ are given by

(3)
$\overline{a}_{ij}=\frac{1}{|Y|}\int_{Y}\left(a_{ij}(y)+a_{ik}(y)\frac{\partial\chi_{j}}{\partial
y_{k}}(y)\right)\,dy,$

where summation over repeated indices is implied, and $\chi_{j}$ solves the
cell problem posed in the unit cell $Y=[-1/2,1/2]^{d}$

(4) $\displaystyle-\nabla\cdot\left(a\nabla\chi_{j}\right)$
$\displaystyle=\nabla\cdot\left(a\,e_{j}\right),\qquad Y$
$\displaystyle\chi_{j}\,\,\,\,$ $\displaystyle Y\text{-periodic and mean
zero,}$

and $e_{j}$ is the standard unit basis vector in $\mathbb{R}^{d}$. With a
slight modification, a similar result holds for locally periodic media. In
$d=1$, the cell problem reduces to a two-point boundary value problem, and it
is easy to show that $\overline{a}$ is given by the harmonic average of $a$

$\overline{a}=\left(\int_{-1/2}^{1/2}\frac{1}{a(s)}\,ds\right)^{-1}.$

In more general settings, the theory of $\Gamma$-convergence provides
sufficient conditions for $u^{\epsilon}$ to limit to some $\overline{u}$;
however, there does not in general exist a closed expression for
$\overline{a}$, and numerical homogenization techniques are required to
approximate the homogenized tensor.

There exist many strategies for the numerical homogenization of (1); examples
include the multiscale finite element method (MsFEM) [37, 36, 24], upscaling
techniques [11], the variational multiscale method [1], the heterogeneous
multiscale method (HMM) [22, 9], the equation-free method [40], and local
orthogonal decomposition (LOD) [42]. Review articles can be found in [33] and
[10]. Each method typically involves solving a microscale problem to estimate
some missing data from the macroscopic model. Because many examples of
realistic heterogeneous media are approximately periodic [18, 28], in
practice, both the size of the microscale domain and the boundary conditions
prescribed for the microscale problem play an important role in the accuracy
of the homogenization procedure.

A common way to study the effect that the choice of microscale domain size and
boundary conditions has on numerical homogenization techniques is the setting
of periodic oscillations, where analytic results are known, but where the
period is assumed to be unknown. In this context, E, Ming, and Zhang [23]
first proved that if the microscale problem is solved on a domain whose length
$\eta$ does not coincide with the true period of the heterogeneous media
$\epsilon$, a boundary, or “cell resonance” error results that is proportional
to $\epsilon/\eta$ and $\epsilon/\eta+\eta$ for truly periodic and locally
periodic data, respectively. Yue and E [44] performed a numerical study of
this boundary error as a function of domain length $\eta$ for Dirichlet,
Neumann, and periodic boundary conditions and found results consistent with
the theory from [23]. Since an $\mathcal{O}(\epsilon/\eta)$ error can
potentially dominate discretization error in any sensible numerical
homogenization scheme, its reduction can be of great practical importance. One
strategy to reduce the cell resonance error is to simply take
$\eta\gg\epsilon$. This, however, will of course greatly increase the overall
computational cost of the homogenization procedure. Instead, it is preferable
to devise strategies to asymptotically reduce the cell resonance error to
$\mathcal{O}(\epsilon/\eta)^{r}$ for some $r>1$, so that greater accuracy can
be obtained at a lower cost.

Several techniques to address the cell resonance error exist in the
literature. Most involve the use of smooth, compactly supported averaging
kernels, or “masks”, when computing the homogenized coefficients; their
compact support helps reduce the influence of the boundary error. The masks
are typically inserted into the “data estimation”, or upscaling step of the
numerical homogenization procedure, that is, the method-specific analog of
(3). In the context of the MsFEM, for example, this is called oversampling
[35, 38], while in HMM it is called filtering [44, 22]. Although these kernels
indeed help reduce the cell resonance error, they only help lower the
prefactor in the error’s rate of decay; the rate itself remains proportional
to $\epsilon/\eta$ [44, 30]. In [16], however, Blanc and Le Bris proposed to
insert the averaging kernels directly in the cell problem (4) itself, in
contract to simply utlizing them as post-processing tools. In general this
technique helped improve the decay rate of the cell resonance error to second
order in $\epsilon/\eta$.

There are a few other notable examples that modify the form of the cell
problem to reduce the influence of the boundary. In [31], Gloria modified the
microscale elliptic problem to include a zeroth order term. Since the Green’s
function associated to the modified elliptic operator decays exponentially
fast, the influence from the boundary is lessened, and a convergence rate of
$\mathcal{O}(\epsilon/\eta)^{4}$ for the cell resonance error was obtained in
the periodic setting. The approach was improved in a follow up work [32] using
Richardson extrapolation. In [13], Arjmand and Runborg proposed to solve a
time-dependent hyperbolic PDE on the microscale, based on the idea that the
finite speed of propagation of the initial data prevents the boundary error
from affecting the solution in interior of the domain, provided the domain is
sufficiently large. Using averaging kernels with special regularity and
vanishing moment properties, they were able to rigorously obtain in the
periodic setting a convergence rate of $\mathcal{O}(\epsilon/\eta)^{q}$ for
arbitrary $q$, depending on the regularity properties of the averaging kernel.
More recently, Abdulle et al. [2, 3, 4] proposed two separate but related
methods to reduce the boundary error. The first approach solves a parabolic
microscale problem over a time interval $T$ that depends on the domain length
$\eta$ and the bounds for the oscillatory tensor $a$ (2). Similar to [31], the
method is based on the exponential decay of the parabolic Green’s function.
The second approach solves an elliptic equation with a modified forcing term
of the form

(5) $f=\sum_{k=1}^{N}e^{-\lambda_{k}T}\left\langle\nabla\cdot
a,\varphi_{k}\right\rangle\varphi_{k},$

where $\\{\lambda_{k},\varphi_{k}\\}_{k=1}^{N}$ are the first $N$ dominant
eigenvalues and eigenfunctions of the operator
$A:=-\nabla\cdot\left(a\nabla\right)$ on $[-\eta/2,\eta/2]^{d}$ equipped with
Dirichlet boundary conditions; this $f$ is a spectral truncation of the
solution operator $e^{-AT}$ for the parabolic problem from semigroup theory.
With appropriate selections for the parameters $T$ and $N$, the authors prove
exponential decay of the boundary error; see also [5] for an extension to the
case of stochastic coefficients.

Rather than modifying the form of the cell problem, we present below an
alternative method for reducing the cell resonance error based on the simple
observation that, as a function of the microscale domain size $\eta$, the
error itself is oscillatory. This was observed in the first numerical study of
Yue and E; see e.g. Fig. 6 in [44]. One goal of the present work is to more
systematically characterize the resonance error as $\eta$ changes. In
dimension $d=1$ we show that the error can be written

(6) $\frac{1}{x}P(x)+R(x),$

where $x=\eta/\epsilon$, $P(x)$ is a periodic function, and $R(x)$ is a
“corrector” term given by

(7) $R(x)=\sum_{k=2}^{\infty}\left(-\frac{1}{x}P(x)\right)^{k}.$

In the case of dimension $d>1$, a decomposition of the form (6) is more
difficult to show, as the resonance error depends on the solution to an
elliptic problem with boundary data that oscillates as a function of $x$. The
oscillatory boundary condition can lead to complicated boundary layers in the
interior; see e.g. [29] and [26]. For two dimensional microscale domains of
the form

(8) $I_{\eta}=[-\eta/2,\eta/2]\times[-\epsilon/2,\epsilon/2],$

we show using Floquet theory [27] that the cell resonance error can also be
written in the form (6), where the corrector $R(x)$ is given by the average of
a locally $\epsilon$-periodic function. More generally for domains of the form
$[-\eta/2,\eta/2]^{d}$ that are relevant to numerical homogenization we offer
numerical evidence that something analogous to (6) indeed holds, but a fully
general theory will likely not be straightforward to develop because of the
aforementioned boundary layers.

Based on the oscillatory nature of the cell resonance error, we also describe
a technique to asymptotically reduce it by taking a weighted average at
several domain sizes. The method is motivated by the form of the one
dimensional corrector (7), but it is designed to work more generally in higher
dimensions. It makes use of averaging kernels that possess regularity
properties and vanishing “negative” moment conditions that accelerate the
error’s convergence to zero at large domain sizes relative to $\epsilon$. The
kernels are similar to, but distinct from, those utilized in [25, 13, 41]. In
dimension $d=1$ we prove the boundary error then decays as

(9) $\mathcal{O}\left(\epsilon/\eta\right)^{\min\\{p+1,q+3\\}}$

where $q$ and $p$ depend on the regularity and vanishing moment properties of
the kernel, respectively. Since the proof only relies on the decomposition
(7), the result would immediately generalize if (7) were shown to hold for
$d>1$. In lieu of a fully general theory, however, we present numerical
examples that suggest the strategy indeed is effective in higher dimensions.

As mentioned above, the proposed method does not modify the form of the
original cell problem from homogenization theory. Instead, it involves solving
a series of microscale problems at different cell sizes $\eta$. Hence, there
is no need to develop additional approximations to other operators, in
contrast to existing methods proposed in the literature. In this regard, the
method is well suited to be combined with existing reduced basis (RB)
techniques in the numerical homogenization literature [6, 8, 17, 43], where
the boundary resonance error is an open problem. Although we do not address
this in the current work, we offer some further discussion in Section 4.3
below.

The remainder of the paper is structured as follows. Section 2 below
introduces the space of averaging kernels $\mathbb{K}^{-p,q}$ and some of its
properties. Since the results are similar to others available in the
literature, their proofs can be found in Appendix A. In Section 3.1 we derive
the expressions (6) and (7) and then use the averaging kernels to prove the
cell resonance error can be asymptotically reduced to (9). After deriving an
expression for the corrector $R$ in (6) for microscale domains of the form (8)
in Section 3.2, a numerical strategy for general situations is described in
Section 3.3. Numerical results are presented and discussed in Section 4, and
then concluding remarks are offered in Section 5.

##  2 Averaging kernels

Let $C(X)$ denote the space of continuous functions on some
$X\subseteq\mathbb{R}$, and let $C^{k}(X)$ be the space of functions that are
$k$ times continuously differentiable. For compact $X$, let $f\in
C_{0}^{k}(X)$ if $f$ and its $k$ derivatives vanish at the endpoints of $X$.

######  Definition 2.1.

For $g\in C(\mathbb{R})$ let $g^{[0]}(x)=g(x)$, and define the primitive

$g^{[k+1]}(x):=\int_{0}^{x}g^{[k]}(s)\,ds+c_{k+1},\qquad k=0,1,\ldots,$

where the constant is chosen so that $\int_{0}^{1}g^{[k+1]}(s)\,ds=0$.

First, recall the following simple result.

######  Lemma 2.2.

Let $g\in C(\mathbb{R})$ be 1-periodic, so that $g(x+1)=g(x)$ $\forall
x\in\mathbb{R}$, and let $g$ be mean zero, so that $\int_{0}^{1}g(x)\,dx=0$.
Then the primitive $g^{[1]}(x)$ defined above is also continuous, 1-periodic,
and bounded.

The result follows from continuity and the mean-zero assumption; see Appendix
A.

Consider next the “incorrect” average of a mean-zero oscillatory function. Let
$f\in C(\mathbb{R})$ be 1-periodic, and fix $\epsilon>0$ and $\eta>\epsilon$
with $\eta/\epsilon\notin\mathbb{N}$. A simple calculation then shows that the
error in taking the incorrect average of length $\eta$ can be bounded as

(10)
$\left|\frac{1}{\eta}\int_{0}^{\eta}f\left(\frac{x}{\epsilon}\right)dx\right|=\left|\frac{\epsilon}{\eta}\int_{0}^{\eta/\epsilon-\lfloor\eta/\epsilon\rfloor}f(u)du\right|\leq
C\,\left\lVert f\right\rVert_{\infty}\left(\frac{\epsilon}{\eta}\right),$

where $C=(\eta/\epsilon-\lfloor\eta/\epsilon\rfloor)\in(0,1)$. Thus, the error
decreases like $\epsilon/\eta$. This error rate can be asymptotically
improved, however, if the average is taken against smooth kernels with nice
regularity properties.

We next define such averaging kernels that additionally possess vanishing
“negative” moments and will be used in the numerical homogenization procedure.
These are similar to other kernels with vanishing “positive” moments found the
literature; see e.g. [25, 12, 13].

######  Definition 2.3.

Let $p$ and $q$ be nonnegative integers. Define
$K\in\mathbb{K}^{-p,q}\left([1,2]\right)$ if $K\in C^{q}_{0}([1,2])$,
$K^{(q+1)}\in BV([1,2])$, and

(11) $\int_{1}^{2}K(x)x^{-r}dx=\begin{cases}1,\qquad r=0\\\ 0,\qquad
r=1,\ldots,p.\end{cases}$

Note that for $K_{\eta}(x):=\frac{1}{\eta}K(x/\eta)$,
$K\in\mathbb{K}^{-p,q}\left([1,2]\right)\iff
K_{\eta}\in\mathbb{K}^{-p,q}\left([\eta,2\eta]\right)$.

The following lemma establishes the bounds needed to prove the main result in
Section 3.1. Since analogous results can be found in the literature for
different kernels,***See, for example, Lemma 2.3 in [12]. the proof is moved
to Appendix A.

######  Lemma 2.4.

Let $K\in\mathbb{K}^{-p,q}\left([1,2]\right)$, and let
$K_{\eta}(x):=\frac{1}{\eta}K(x/\eta)$ for $\eta>0$. Let $f\in C(\mathbb{R})$
be 1-periodic, and let $\left\langle f\right\rangle=\int_{0}^{1}f(s)ds$. Let
$0<\epsilon<\eta$, and for $r\in\mathbb{N}$ let

$\Upsilon_{r}(\eta):=\int_{\eta}^{2\eta}K_{\eta}(x)x^{-r}f\left(\frac{x}{\epsilon}\right)\,dx.$

Then:

(12) $\left|\Upsilon_{r}(\eta)\right|\leq\begin{cases}C_{1}\,\left\lVert
f\right\rVert_{\infty}r^{q+1}\,\left(\frac{\epsilon}{\eta}\right)^{q+2}\eta^{-r},\qquad
r\leq p\\\ C_{1}\,\left\lVert
f\right\rVert_{\infty}r^{q+1}\,\left(\frac{\epsilon}{\eta}\right)^{q+2}\eta^{-r}+\left|\left\langle
f\right\rangle\right|\left\lVert K\right\rVert_{\infty}\eta^{-r},\qquad
r>p\end{cases}$

where $C_{1}$ is independent of $\epsilon$, $\eta$, and $r$, and depends only
on $q$, $K$ and its derivatives.

For fixed $\epsilon$, Figure 1 compares the decay as a function of $\eta$ of
$\eta^{-r}f(\eta/\epsilon)$ and $\Upsilon_{r}(\eta)$ for two different
periodic functions $f$; the rates in Lemma 2.4 are seen to be sharp. The
averaging kernels $K\in\mathbb{K}^{-p,q}$ used are of the form

(13) $K(x)=(x-1)^{q+1}(x-2)^{q+1}\sum_{j=0}^{p}a_{j}x^{j},$

where the coefficients $\\{a_{j}\\}_{j=0}^{p}$ are chosen to ensure (11)
holds.

Figure 1:  The decay of $\eta^{r}f(\eta/\epsilon)$ and $\Upsilon_{r}(\eta)$
versus $\eta$ for various averaging kernels $K\in\mathbb{K}^{-p,q}$ of the
form (13). (Left) $r=2$ and $f(x)=1.1+\sin(2\pi x)$. (Right) $r=1$ and $f(x)$
is the even periodic extension of the piecewise constant function
$\chi_{[0,1/4]}+\chi_{[1/4,1/2]}$.

Having shown the requisite facts for the function class $\mathbb{K}^{-p,q}$,
we are now ready to develop our strategy for reducing the boundary error.

##  3 Reducing the cell resonance error

###  3.1 One dimension

Here and throughout the section, let $a:\mathbb{R}\to\mathbb{R}$ be a
1-periodic, continuous function that satisfies

$0<\lambda<a(s)<\Lambda,\qquad\forall s\in\mathbb{R}.$

In one dimension the cell problem (4) rescaled by $\epsilon$ is

(14) $\displaystyle-\frac{d}{dx}\Big{(}a^{\epsilon}(x)$
$\displaystyle\frac{d\chi}{dx}(x)\Big{)}=\frac{d}{dx}a^{\epsilon}(x),\qquad
x\in[-\epsilon/2,\epsilon/2]$ $\displaystyle\chi\,\,\,\,$
$\displaystyle\epsilon\text{-periodic and mean zero}$

with corresponding homogenized coefficient

(15)
$\overline{a}=\left(\frac{1}{\epsilon}\int_{-\epsilon/2}^{\epsilon/2}\frac{1}{a^{\epsilon}(s)}ds\right)^{-1},$

where $a^{\epsilon}(x)=a(x/\epsilon)$. To study the cell resonance error we
consider the boundary value problem (14) posed instead on the domain
$I_{\eta}=[-\eta/2,\eta/2]$, where $\eta$ is inconsistent with the periodicity
$\epsilon$ of the coefficient $a^{\epsilon}$. The corresponding “incorrect”
harmonic average is then

(16)
$\left(\frac{1}{\eta}\int_{-\eta/2}^{\eta/2}\frac{1}{a^{\epsilon}(s)}ds\right)^{-1}.$

After rescaling the integral by $\epsilon$, the periodic integrand $1/a(s)$
can be decomposed into the sum of odd and even contributions, and since the
odd part will vanish by symmetry, (16) becomes

$\left(\frac{2\epsilon}{\eta}\int_{0}^{\eta/2\epsilon}b_{\rm
even}(s)ds\right)^{-1},$

where $b_{\rm even}$ denotes the even part of $b(s):=1/a(s)$. In the
development below, we hence assume without loss of generality that $b$ is an
even function, and for $\eta>\epsilon/2$ we define the “incorrect” harmonic
average

(17)
$\rho^{\epsilon}(\eta):=\left(\frac{\epsilon}{\eta}\int_{0}^{\eta/\epsilon}b(s)\,ds\right)^{-1}.$

After a simple algebraic rearrangement, a calculation similar to (10) gives

$\left|\overline{a}-\rho^{\epsilon}(\eta)\right|\leq
C\left(\frac{\epsilon}{\eta}\right),$

which is the one-dimensional version of the cell-resonance error first derived
in [23]. Within this bound, however, the error has a richer structure, which
we show next.

######  Lemma 3.1.

Let

(18)
$B=\left\lVert\frac{1}{a}-\int_{0}^{1}\frac{1}{a(s)}ds\right\rVert_{L^{\infty}(\mathbb{R})}$

be the supremum of the fluctuations about the mean for the function $1/a$, and
let $\overline{a}$ be the true harmonic average (15). If
$\eta/\epsilon>\overline{a}B$, then $\rho^{\epsilon}$ is given by the
expansion

(19)
$\rho^{\epsilon}(\eta)=\overline{a}\sum_{k=0}^{\infty}\left(-\frac{\epsilon}{\eta}\,q(\eta/\epsilon)\right)^{k}$

where $q(\eta/\epsilon)$ is an $\epsilon$-periodic function of $\eta$.

Proof: Let $b(s):=1/a(s)$, and let $\left\langle
b\right\rangle:=\int_{0}^{1}b(s)\,ds$ be its arithmetic average. Note
$\overline{a}=1/\left\langle b\right\rangle$. Then

$\displaystyle\rho^{\epsilon}(\eta)$ $\displaystyle=\left(\left\langle
b\right\rangle+\frac{\epsilon}{\eta}\int_{0}^{\eta/\epsilon}\left(b(s)-\left\langle
b\right\rangle\right)\,ds\right)^{-1}$
$\displaystyle=\overline{a}\left(1+\overline{a}\,\,\frac{\epsilon}{\eta}\int_{0}^{\eta/\epsilon}\left(b(s)-\left\langle
b\right\rangle\right)\,ds\right)^{-1}$
$\displaystyle=\overline{a}\left(1+z\left(\eta/\epsilon\right)\right)^{-1},$

where

$z(\eta/\epsilon)=\overline{a}\,\,\frac{\epsilon}{\eta}\int_{0}^{\eta/\epsilon}\left(b(s)-\left\langle
b\right\rangle\right)ds.$

The condition $\eta/\epsilon>\overline{a}B$ ensures that
$|z(\eta/\epsilon)|<1$ and hence that $\rho^{\epsilon}$ can be expanded in a
uniformly convergent geometric series, as desired. Since $b(s)-\left\langle
b\right\rangle$ is a mean-zero, periodic function,

$q(\eta/\epsilon):=z(\eta/\epsilon)\frac{\eta}{\epsilon}=\overline{a}\int_{0}^{\eta/\epsilon}(b(s)-\left\langle
b\right\rangle)ds$

is also $\epsilon$-periodic by Lemma 2.2. $\square$

The decomposition of the cell resonance error (19) motivates the use of
weighted averages against kernels $K\in\mathbb{K}^{-p,q}$, which we define
next.

######  Definition 3.2.

For $K\in\mathbb{K}^{-p,q}\left([1,2]\right)$, $\eta>0$ and
$K_{\eta}(x)=\frac{1}{\eta}K(x/\eta)$, define

(20)
$S^{\epsilon}(\eta):=\int_{\eta}^{2\eta}K_{\eta}(x)\rho^{\epsilon}(x)\,dx.$

The averaged error (20) accelerates the convergence of $\rho^{\epsilon}(\eta)$
to the true harmonic average $\overline{a}$ at a rate that depends on $p$ and
$q$, which we now show.

######  Theorem 3.3.

For non-negative integers $p$ and $q$ let
$K\in\mathbb{K}^{-p,q}\left([1,2]\right)$, and let $\epsilon,\eta>0$ such that

(21) $\overline{a}B\left(1+\frac{1}{p+1}\right)^{q+1}<\frac{\eta}{\epsilon}$

where $B$ is given by (18) and $\overline{a}$ is the true harmonic average
(15). Let $S^{\epsilon}$ be defined by (20). Then

(22) $\left|\,S^{\epsilon}(\eta)-\overline{a}\,\right|\leq
C\,\varphi\big{(}\overline{a}B\frac{\epsilon}{\eta}\big{)}\left(\frac{\epsilon}{\eta}\right)^{\min\\{p+1,q+3\\}},$

where $C$ depends on $p$, $q$, $K$, $\overline{a}$ and $B$, and
$\varphi(\theta):=1/(1-\theta)$.

Proof: Since the series expansion (19) converges uniformly and absolutely, it
can be inserted into the weighted average (20), and order of the sum and
integral can be interchanged. The resulting expression is

$\displaystyle S^{\epsilon}(\eta)$
$\displaystyle=\int_{\eta}^{2\eta}K_{\eta}(x)\rho^{\epsilon}(x)dx$
$\displaystyle=\overline{a}+\sum_{r=1}^{\infty}(-1)^{r}\overline{a}^{r+1}\epsilon^{r}\int_{\eta}^{2\eta}\Big{[}K_{\eta}(x)x^{-r}\overbrace{\left(\int_{0}^{x/\epsilon}\left(b(s)-\left\langle
b\right\rangle\right)ds\right)^{r}\,}^{G_{r}(x/\epsilon):=}\,\Big{]}dx,$

which implies

(23)
$\left|S^{\epsilon}(\eta)-\overline{a}\right|\leq\sum_{r=1}^{\infty}\overbrace{\overline{a}^{r+1}\epsilon^{r}\left|\int_{\eta}^{2\eta}K_{\eta}(x)x^{-r}G_{r}(x/\epsilon)dx\right|}^{F_{r}:=}.$

Fixing $r$, each individual term in the summations can be bounded using Lemma
2.4. Since $b-\left\langle b\right\rangle$ periodic, and mean-zero, and
continuous, Lemma 2.2 implies $G_{r}$ is both periodic and continuous as well.
Hence, the assumptions of Lemma 2.4 are indeed satisfied, so that

$\displaystyle F_{r}$
$\displaystyle\leq\overline{a}^{r+1}\epsilon^{r}\cdot\begin{cases}C(q,K)\left\lVert
G_{r}\right\rVert_{\infty}\,r^{q+1}\left(\frac{\epsilon}{\eta}\right)^{q+2}\eta^{-r},\qquad
r\leq p\\\ C(q,K)\left\lVert
G_{r}\right\rVert_{\infty}\,r^{q+1}\left(\frac{\epsilon}{\eta}\right)^{q+2}\eta^{-r}+\left|\left\langle
G_{r}\right\rangle\right|\left\lVert K\right\rVert_{\infty}\eta^{-r},\qquad
r>p\end{cases}$ (24) $\displaystyle=\begin{cases}C(q,K)\left\lVert
G_{r}\right\rVert_{\infty}r^{q+1}\,\overline{a}^{r+1}\left(\frac{\epsilon}{\eta}\right)^{r+q+2},\qquad
r\leq p\\\ C(q,K)\left\lVert
G_{r}\right\rVert_{\infty}\,r^{q+1}\,\overline{a}^{r+1}\left(\frac{\epsilon}{\eta}\right)^{r+q+2}+\left|\left\langle
G_{r}\right\rangle\right|\left\lVert
K\right\rVert_{\infty}\,\overline{a}^{r+1}\left(\frac{\epsilon}{\eta}\right)^{r},\qquad
r>p.\end{cases}$

Furthermore, because $b-\left\langle b\right\rangle$ is periodic and mean
zero,

(25) $\sup_{x\in\mathbb{R}}\left|\int_{0}^{x/\epsilon}(b(s)-\left\langle
b\right\rangle)\,ds\right|\leq B,$

and hence

(26) $\left\lVert G_{r}\right\rVert_{\infty}\leq B^{r};$

an identical bound exists for $|\left\langle G_{r}\right\rangle|$. The
infinite sum (23) can be broken up into two pieces, at which point the bounds
(24) and (26) can be inserted to bound the difference between the true
harmonic average and $S^{\epsilon}(\eta)$ by three parts:

$\displaystyle\left|S^{\epsilon}(\eta)-\overline{a}\right|\leq\sum_{r=1}^{p}$
$\displaystyle
C(q,K)r^{q+1}\overline{a}^{r+1}B^{r}\left(\frac{\epsilon}{\eta}\right)^{r+q+2}$
(27) $\displaystyle+$
$\displaystyle\sum_{r=p+1}^{\infty}\Big{[}C(q,K)r^{q+1}\overline{a}^{r+1}B^{r}\left(\frac{\epsilon}{\eta}\right)^{r+q+2}+\left\lVert
K\right\rVert_{\infty}\overline{a}^{r+1}B^{r}\left(\frac{\epsilon}{\eta}\right)^{r}\Big{]},$

The first part on the right hand side (RHS) of the inequality (27) is a finite
sum, which can be bounded as

(28)
$\sum_{r=1}^{p}C(q,K)r^{q+1}\overline{a}^{r+1}B^{r}\left(\frac{\epsilon}{\eta}\right)^{r+q+2}\leq
C(q,K)\,\overline{a}^{2}B\,p^{q+2}\left(\frac{\epsilon}{\eta}\right)^{q+3}.$

The third part of the RHS of (27) is the tail of a geometric series. The
assumption (21) implies

(29) $\sum_{r=p+1}^{\infty}\left\lVert
K\right\rVert_{\infty}\overline{a}\,(\overline{a}B)^{r}\left(\frac{\epsilon}{\eta}\right)^{r}=\left\lVert
K\right\rVert_{\infty}\overline{a}\,(\overline{a}B)^{p+1}\left(\frac{\epsilon}{\eta}\right)^{p+1}\varphi\big{(}\overline{a}B\frac{\epsilon}{\eta}\big{)}.$

Finally, the second part of the RHS of (27) can be bounded with the “integral
test” from elementary calculus, which says

$\sum_{r=p+1}^{\infty}f(r)\leq f(p+1)+\int_{p+1}^{\infty}f(r)\,dr,$

provided $f$ is a monotonically decreasing function of $r$. Here
$f(r)=r^{q+1}(\overline{a}B)^{r}\left(\frac{\epsilon}{\eta}\right)^{r}$, and
the assumption (21) guarantees that $f(r+1)<f(r)$ indeed holds for every
$r>p$. Hence

$\displaystyle\sum_{r=p+1}^{\infty}C(q,K)r^{q+1}(\overline{a}B)^{r}\left(\frac{\epsilon}{\eta}\right)^{r+q+2}\leq
C(q,K)$
$\displaystyle\Big{[}(p+1)^{q+1}(\overline{a}B)^{p+1}\left(\frac{\epsilon}{\eta}\right)^{p+q+3}$
(30)
$\displaystyle+\left(\frac{\epsilon}{\eta}\right)^{q+2}\underbrace{\int_{p+1}^{\infty}r^{q+1}\alpha^{r}\,dr}_{(\ast)}\Big{]},$

where $\alpha=\overline{a}B\epsilon/\eta$. The integral $(\ast)$ can be
computed with $q+1$ applications of integration by parts:

$\displaystyle(\ast)$
$\displaystyle=-\int_{p+1}^{\infty}(q+1)r^{q}\left(\log(\alpha)\right)^{-1}\alpha^{r}dr+(p+1)^{q+1}\left(-\log(\alpha)\right)^{-1}\alpha^{p+1}$
$\displaystyle=\int_{p+1}^{\infty}(q+1)!\left(-\log(\alpha)\right)^{-(q+1)}\alpha^{r}dr$
$\displaystyle+\sum_{k=1}^{q+1}\frac{(q+1)!}{(q+2-k)!}(p+1)^{q+2-k}\left(-\log(\alpha)\right)^{-k}\alpha^{p+1}$
(31) $\displaystyle=(q+1)!\left(-\log(\alpha)\right)^{-(q+2)}\alpha^{p+1}$
$\displaystyle+\alpha^{p+1}\sum_{k=1}^{q+1}\frac{(q+1)!}{(q+2-k)!}(p+1)^{q+2-k}\left(-\log(\alpha)\right)^{-k}$

Comparing the dependence on $\epsilon/\eta$ in the upper bounds on the three
parts on the RHS of the inequality (27) (given in (28), (29), and both (30)
and (31)) then gives the desired result:

$\left|S^{\epsilon}(\eta)-\overline{a}\right|\leq
C(p,q,K,\overline{a},B)\varphi\big{(}\overline{a}B\frac{\epsilon}{\eta}\big{)}\left(\frac{\epsilon}{\eta}\right)^{\min\\{p+1,q+3\\}}\square$

Next we note that in the special case that $K(x)=1\in\mathbb{K}^{0,-1}$, (20)
becomes a simple arithmetic average of $\rho^{\epsilon}$, and the result (22)
implies that the averaged error asymptotically decays as $\epsilon/\eta$. A
more direct calculation, however, reveals that this estimate can be improved
to second order in $\epsilon/\eta$.

######  Theorem 3.4.

Let $\eta/\epsilon>\overline{a}B$. Then

(32)
$\left|\frac{1}{\eta}\int_{\eta}^{2\eta}(\rho^{\epsilon}(x)-\overline{a})\,dx\right|\leq
C\,\varphi\big{(}\overline{a}B\frac{\epsilon}{\eta}\big{)}\left(\frac{\epsilon}{\eta}\right)^{2}$

where $C$ depends on $\overline{a}$ and $B$, and
$\varphi(\theta):=1/(1-\theta)$.

Since the proof stategy is similar to that of Theorem 3.3, it is presented in
Appendix B.

In practice, we find in the numerical experiments presented in Section 4 that
$\left\lVert K\right\rVert_{\infty}$ grows relatively fast as $p$ and $q$
grow, and hence the error decay rate predicted by (22) is not seen until
$\eta/\epsilon$ is relatively large. Although the decay rate (32) is only
second order in $\epsilon/\eta$, this relatively simple averaging strategy is
found to be most effective in practice, particularly in higher dimensions
where solving the cell problem (4) on a domain with size
$\delta:=\eta/\epsilon$ in each direction becomes computationally expensive at
large $\delta$.

###  3.2 Tubular domains in two dimensions

Next, let dimension $d=2$, and in the following assume that $a(x_{1},x_{2})$
is an isotropic, coercive tensor that is $1$-periodic in both arguments.
Because we make use of classical Sturm-Liouville theory below, we additionally
assume that $a$ is once continuously differentiable.

For $\delta=\eta/\epsilon$, consider domains of the form

$I_{\delta}=[-\delta/2,\delta/2]\times[-1/2,-1/2],$

as depicted in Figure 2. Assume $\delta\notin\mathbb{N}$, so that there is a
mismatch between the size of the micro domain and the periodicity of $a$ in
the $x_{1}$ direction.

$I_{\delta}$$1$$x_{2}$$x_{1}$$\delta$ Figure 2: A two-dimensional “tubular”
domain $I_{\delta}=[-\delta/2,\delta/2]\times[-1/2,1/2]$,
$\delta\notin\mathbb{N}$.

We show that the cell resonance error can be written in the form (6) from the
introduction, where the “corrector” $R(\delta)$ is given by the average of a
locally $\epsilon$-periodic function.

Consider first the periodic extension to all of $\mathbb{R}^{2}$ of the
solution $\chi_{j}$, $j\in\\{1,2\\}$, to the true cell problem (4) (i.e. with
$\delta=1$). Next consider the solution $\phi_{\delta,j}$ to the cell problem
posed on the “incorrect” domain $I_{\delta}$:

$\displaystyle-\nabla\cdot\big{(}a\nabla\phi_{\delta,j}\big{)}$
$\displaystyle=\nabla\cdot\big{(}ae_{j}\big{)},\qquad I_{\delta}$ (33)
$\displaystyle\int_{I_{\delta}}\phi_{\delta,j}\,dx$ $\displaystyle=0$

with periodic boundary conditions, and define the difference

(34)
$u_{\delta,j}(x_{1},x_{2}):=\phi_{\delta,j}(x_{1},x_{2})-\chi_{j}(x_{1},x_{2}).$

Note that if $\delta\in\mathbb{N}$, then $u_{\delta,j}=0$ identically. The
standard elliptic theory guarantees that both $\chi_{j}$ and
$\phi_{\delta,j}$, (and hence $u_{\delta,j}$) are twice continuously
differentiable. More generally $u_{\delta,j}$ solves the following elliptic
problem in $I_{\delta}$:

$\displaystyle-\nabla\cdot\big{(}a\nabla u_{\delta,j}\big{)}=0$
$\displaystyle,\qquad I_{\delta}$ $\displaystyle
u_{\delta,j}(x_{1},-1/2)=u_{\delta,j}(x_{1},1/2)$ $\displaystyle,\qquad
x_{1}\in[-\delta/2,\delta/2]$ $\displaystyle u_{\delta,j}=g_{\delta,j}(x_{2})$
$\displaystyle,\qquad x_{1}=-\delta/2$ (35) $\displaystyle
u_{\delta,j}=h_{\delta,j}(x_{2})$ $\displaystyle,\qquad x_{1}=\delta/2$

where $g_{\delta,j}$ and $h_{\delta,j}$ are simply the difference between
$\phi_{\delta,j}$ and $\chi_{j}$ at the points $x_{1}=\mp\delta/2$. Since
$\delta$ is assumed to be inconsistent with the periodicity of $a$, both
$g_{j}$ and $h_{j}$ are oscillatory functions of $\delta$ for fixed $x_{2}$.
Both $u_{\delta,j}$ and $\phi_{\delta,j}$ are of course $1$-periodic in
$x_{2}$; we next show that they are also both locally $1$-periodic in $x_{1}$.

######  Theorem 3.5.

The solution $\phi_{\delta,j}(x_{1},x_{2})$ to the elliptic problem (33) is
locally $1$-periodic in $x_{1}$.

Proof: Because $\chi_{j}$ is $1$-periodic, it is sufficient to show that the
difference $u_{\delta,j}=\phi_{\delta,j}-\chi$ is locally $1$-periodic. Since
the proof proceeds identically for $j\in\\{1,2\\}$, we drop the subscript
below for notational convenience.

By assumption of isotropy, the governing equation for $u_{\delta}$ simplifies
to

(36) $-\left(\frac{\partial}{\partial
x_{1}}\Big{(}a(x_{1},x_{2})\frac{\partial}{\partial
x_{1}}\Big{)}+\frac{\partial}{\partial
x_{2}}\Big{(}a(x_{1},x_{2})\frac{\partial}{\partial
x_{2}}\Big{)}\right)u_{\delta}(x_{1},x_{2})=0.$

For fixed $x_{1}\in\mathbb{R}$, define the symmetric, positive-definite
differential operator $L(x_{1})$ by:

$L(x_{1})f:=-\frac{\partial}{\partial
x_{2}}\left(a(x_{1},x_{2})\frac{\partial}{\partial x_{2}}f\right),$

acting on $C^{2}$ functions $f:[-1/2,1/2]\to\mathbb{R}$ with periodic boundary
conditions. Sturm-Liouville theory [20] guarantees that $L(x_{1})$ has a
countably infinite set of eigenvalues
$\\{\lambda_{i}(x_{1})\\}_{i=0}^{\infty}$ that satisfy $\lambda_{0}(x_{1})=0$
and $\lambda_{i+1}(x_{1})>\lambda_{i}(x_{1})$ for all $i\geq 0$, as well as an
associated set of complete, mutually orthonormal eigenfunctions
$\\{\varphi_{i}(x_{1},x_{2})\\}_{i=0}^{\infty}$. Note also that by periodicity
of $a$, $L(x_{1}+1)=L(x_{1})$ for any $x_{1}\in\mathbb{R}$. Since the
eigenvalues and their orthonormal eigenfunctions are unique, this implies that
both are $1$-periodic in $x_{1}$.

Consider an eigenfunction expansion of the solution to the boundary value
problem (BVP) (35)

(37)
$u_{\delta}(x_{1},x_{2})=\sum_{i=0}^{\infty}\alpha_{i}(x_{1})\varphi_{i}(x_{1},x_{2})$

where

$\alpha_{i}(x_{1})=\int_{-1/2}^{1/2}u_{\delta}(x_{1},x_{2})\varphi_{i}(x_{1},x_{2})\,dx_{2}.$

Since $u_{\delta}$ is $C^{2}$, the expansion converges uniformly and
absolutely.

Inserting (37) into (36) then gives

$\sum_{i=0}^{\infty}\Big{[}-\frac{\partial}{\partial
x_{1}}\Big{(}a(x_{1},x_{2})\frac{\partial}{\partial
x_{1}}\big{(}\alpha_{i}(x_{1})\varphi_{i}(x_{1},x_{2})\big{)}\Big{)}+\lambda_{i}(x_{1})\alpha_{i}(x_{1})\varphi_{i}(x_{1},x_{2})\Big{]}=0$

Let $\psi_{i}(x_{1},x_{2}):=\alpha_{i}(x_{1})\varphi_{i}(x_{1},x_{2})$. A
sufficient condition for $u$ to solve (36) is for each $\psi_{i}$ to satisfy
the following boundary value problem (BVP) in $x_{1}$ that is parameterized by
$x_{2}\in[-1/2,1/2]$:

$\displaystyle-\frac{\partial}{\partial
x_{1}}\Big{(}a(x_{1},x_{2})\frac{\partial}{\partial
x_{1}}\big{(}\psi_{i}(x_{1},x_{2})\big{)}\Big{)}+\lambda_{i}(x_{1})\psi_{i}(x_{1},x_{2})=0$
$\displaystyle,\qquad x_{1}\in[-\delta/2,\delta/2]$
$\displaystyle\psi_{i}=\beta_{i}(-\delta/2)\varphi_{i}(-\delta/2,x_{2})$
$\displaystyle,\qquad x_{1}=-\delta/2$ (38)
$\displaystyle\psi_{i}=\gamma_{i}(\delta/2)\varphi_{i}(\delta/2,x_{2})$
$\displaystyle,\qquad x_{1}=\delta/2.$

Here $\beta_{i}(-\delta/2)$ is the $i$-th coefficient in the expansion in
eigenfunctions of $L(-\delta/2)$ of the boundary condition $g_{\delta}(x_{2})$
in (35), while $\gamma_{i}(\delta/2)$ is the coefficient of the expansion of
$h_{\delta}(x_{2})$ in eigenfunctions of $L(\delta/2)$. Since
$\lambda_{i}(x_{1})\geq 0$ for all $i\geq 0$ and $\forall x_{1}\in\mathbb{R}$,
(38) has a unique solution $\forall x_{2}\in[-1/2,1/2]$ and $i\geq 0$.

For each solution $\psi_{i}(x_{1},x_{2})$, define the “flux”

$v_{i}(x_{1},x_{2}):=a(x_{1},x_{2})\frac{\partial}{\partial
x_{1}}\psi_{i}(x_{1},x_{2})$

and consider a reformulation of the BVP (38) into a linear initial value
problem system

$\frac{\partial}{\partial x_{1}}\begin{pmatrix}v_{i}(x_{1},x_{2})\\\
\psi_{i}(x_{1},x_{2})\end{pmatrix}=\begin{pmatrix}0&\lambda_{i}(x_{1},x_{2})\\\
1/a(x_{1},x_{2})&0\\\ \end{pmatrix}\begin{pmatrix}v_{i}(x_{1},x_{2})\\\
\psi_{i}(x_{1},x_{2})\end{pmatrix}$

for $x_{1}>-\delta/2$, where the initial conditions are simply the values of
$\psi_{i}$ and $v_{i}$ at $x_{1}=-\delta/2$. Since both $1/a$ and
$\lambda_{i}$ are $1$-periodic functions of $x_{1}$, Floquet’s theorem [27]
implies that the solution can be written as

$\begin{pmatrix}v_{i}(x_{1},x_{2})\\\
\psi_{i}(x_{1},x_{2})\end{pmatrix}=P_{i}(x_{1},x_{2})e^{x_{1}K_{i}(x_{2})}c_{i}$

for some $c_{i}\in\mathbb{R}^{2}$. Here $K_{i}$ and $P_{i}$ are both $2\times
2$ matrices that generally depend on $x_{2}$; however, $K_{i}$ is constant in
$x_{1}$ and $P_{i}$ is $1$-periodic in $x_{1}$. Hence, each $\psi_{i}$ is
locally $1$-periodic in $x_{1}$. Since the eigenfunction expansion (37)
converges uniformly and absolutely, the infinite sum
$u_{\delta}=\sum_{i=0}^{\infty}\psi_{i}$ is also locally $1$-periodic in
$x_{1}$, as desired. $\square$

######  Remark 3.6.

Note that while Theorem 3.5 characterizes the behavior of
$u_{\delta,j}(x_{1},x_{2})$ (and hence $\phi_{\delta,j}(x_{1},x_{2})$) as a
function of $x_{1}$ for _fixed_ $\delta$, it does not describe the functional
dependence on $\delta$. In general the dependence is felt through the
oscillatory boundary conditions $g_{\delta,j}$ and $h_{\delta,j}$ in (35).

Next, define

$\rho(\delta):=\frac{1}{\delta}\int_{-\delta/2}^{\delta/2}\int_{-1/2}^{1/2}(a+a\nabla\phi_{\delta,j})\,dx_{2}dx_{1}$

to be the approximation to the homogenized tensor with cell size $\delta$; the
true homogenized coefficients $\overline{a}$ result when $\delta=1$, or more
generally when $\delta\in\mathbb{N}$. As a corollary to Theorem 3.5, we next
show the cell resonance error can be written in the form (6) from the
introduction.

######  Corollary 3.7.

Let $E(\delta)=\rho(\delta)-\overline{a}$ be the cell resonance error. Then

$E(\delta)=\frac{1}{\delta}P(\delta)+\frac{1}{\delta}\int_{-\delta/2}^{\delta/2}\zeta(x_{1};\delta)\,dx_{1}$

where $P$ is an isotropic tensor that is $1$-periodic in $\delta$ and $\zeta$
is a diagonal tensor that is locally $1$-periodic in $x_{1}$.

Proof: First note that since $a$ is isotropic, both $\overline{a}$ and
$\rho(\delta)$ are diagonal: $\overline{a}_{ij}=\rho_{ij}(\delta)=0$ for
$i\neq j$. Next, define for $\nu\in\\{1,2\\}$ the arithmetic averages

$\left\langle
a_{\nu\nu}\right\rangle=\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}a_{\nu\nu}\,dx_{2}dx_{1}$

and

$\left\langle a_{\nu\nu}\frac{\partial\chi_{\nu}}{\partial
x_{\nu}}\right\rangle=\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}a_{\nu\nu}\frac{\partial\chi_{\nu}}{\partial
x_{\nu}}\,dx_{2}dx_{1}$

(here Einstein summation is not implied), so that
$\overline{a}_{\nu\nu}=\left\langle a_{\nu\nu}\right\rangle+\left\langle
a_{\nu\nu}\partial\chi_{\nu}/\partial x_{\nu}\right\rangle$. Using
$\phi_{\delta,\nu}=u_{\delta,\nu}+\chi_{\nu}$ from (34), the resonance error
is given by

(39)
$E_{\nu\nu}(\delta)=\frac{1}{\delta}\int_{-\delta/2}^{\delta/2}\int_{-1/2}^{1/2}\Big{(}p_{\nu}(x_{1},x_{2})+a_{\nu\nu}\frac{\partial
u_{\delta,\nu}}{\partial x_{\nu}}(x_{1},x_{2})\Big{)}dx_{2}dx_{1}$

where

$p_{\nu}(x_{1},x_{2})):=\big{(}a_{\nu\nu}-\left\langle
a_{\nu\nu}\right\rangle\big{)}+\Big{(}a_{\nu\nu}\frac{\partial\chi_{\nu}}{\partial
x_{\nu}}-\left\langle a_{\nu\nu}\frac{\partial\chi_{\nu}}{\partial
x_{\nu}}\right\rangle\Big{)}$

is a mean-zero, $1$-periodic function of both $x_{1}$ and $x_{2}$. Let

$\widetilde{p}_{\nu}(x_{1}):=\int_{-1/2}^{1/2}p_{\nu}(x_{1},x_{2})\,dx_{2},$

and break $\widetilde{p}_{\nu}$ into even and odd parts. Then the first part
of (39) is

$\frac{1}{\delta}\int_{-\delta/2}^{\delta/2}\int_{-1/2}^{1/2}p_{\nu}(x_{1},x_{2})\,dx_{2}dx_{1}=\frac{1}{\delta}\,2\int^{\delta/2}_{0}\widetilde{p}_{\nu,\rm
even}(x_{1})\,dx_{1}=:\frac{1}{\delta}P(\delta),$

and Lemma 2.2 guarantees that the primitive $P(\delta)$ is $1$-periodic.

The second term in the resonance error (39) is

$R(\delta):=\frac{1}{\delta}\int_{-\delta/2}^{\delta/2}\int_{-1/2}^{1/2}a_{\nu\nu}(x_{1},x_{2})\frac{\partial
u_{\delta,\nu}}{\partial x_{\nu}}(x_{1},x_{2})\,dx_{2}dx_{1}.$

By Theorem 3.5, $u_{\nu}$ is locally $1$-periodic in $x_{1}$ for each
$\delta$. Since the gradient $\nabla u_{\nu}$ is also locally $1$-periodic,
setting

$\zeta_{\nu\nu}(x_{1};\delta):=\int_{-1/2}^{1/2}a_{\nu\nu}(x_{1},x_{2})\frac{\partial
u_{\delta,\nu}}{\partial x_{\nu}}(x_{1},x_{2})\,dx_{2}$

for $\nu\in\\{1,2\\}$ gives the desired result. $\square$

###  3.3 General numerical strategy

Motivated by the idea that the cell resonance error is an oscillatory function
of domain size, we now propose a numerical strategy for reducing the boundary
error in general, higher dimensional settings.

Let the dimension $d>1$, and for $\eta>0$, let
$I_{\eta}=[-\eta/2,\eta/2]^{d}$. Let
$a^{\epsilon}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ to be a bounded, coercive
function with entries that are supported at frequencies $\sim 1/\epsilon$. In
general $a^{\epsilon}$ need not be periodic. For $1\leq j\leq d$, let
$\chi_{\eta,j}$ be the solution to the elliptic problem

$\displaystyle-\nabla\cdot\Big{(}a^{\epsilon}(x)\nabla\chi_{\eta,j}(x)\Big{)}$
$\displaystyle=\nabla\cdot(a^{\epsilon}(x)e_{j}),\qquad x\in I_{\eta}$ (40)
$\displaystyle\int_{I_{\eta}}\chi_{\eta,j}\,dx$ $\displaystyle=0$

posed with periodic boundary conditions. Other boundary conditions such as
Dirichlet or Neumann are possible; however, from [44] it is known that
periodic conditions introduce the least boundary effects. In general we assume
(40) possesses a suitably defined weak solution, but since we are ultimately
interested in numerical solutions, we do not further consider any issues of
regularity.

Define for $1\leq i,j\leq d$ the estimate for the homogenized tensor

(41)
$\rho_{ij}(\eta):=\frac{1}{|I_{\eta}|}\int_{I_{\eta}}\Big{(}a_{ij}(x)+a_{ik}\frac{\partial\chi_{\eta,j}}{\partial
x_{k}}(x)\Big{)}dx,$

where summation over $k$ is implied. Note that if $a^{\epsilon}$ is
$\epsilon$-periodic, for example, the true homogenized coefficient
$\overline{a}_{ij}$ is simply given by (41) for $\eta=\epsilon$, while if
$a^{\epsilon}$ is quasi-periodic or stationary ergodic, then
$\overline{a}_{ij}$ is given by the limit as $\eta\to\infty$ of (41), where
$\chi_{\eta,j}$ is replaced by a function $\chi_{\infty,j}$ that solves the
elliptic problem (40) posed on $\mathbb{R}^{d}$ [31, 39].

To improve the approximation of the true homogenized coefficients, we
introduce the averaged coefficients

(42) $S_{ij}(\eta):=\int_{\eta}^{2\eta}K_{\eta}(t)\rho_{ij}(t)\,dt,$

where $K_{\eta}\in\mathbb{K}^{-p,q}([\eta,2\eta])$ for some nonnegative
integers $p$ and $q$. In practice, this integral is approximated by some
quadrature rule defined by $M$ weights $\\{\omega_{m}\\}_{m=1}^{M}$ and nodes
$T_{M}:=\\{t_{m}\\}_{m=1}^{M}\subset[\eta,2\eta]$.

With these preliminary definitions, we now summarize how the smoothed
approximation (42) of the true homogenized coefficients are obtained in
practice.

  1. (1)

For each $t_{m}\in T_{M}$:

    1. (i)

Solve the elliptic problem (40) with $\eta=t_{m}$ for each $1\leq j\leq d$
using e.g. a finite element method, finite difference method, etc.

    2. (ii)

Using a quadrature method on $I_{t_{m}}$ and a suitable estimate for the
gradient of $\chi_{t_{m},j}$, numerically compute and store $\rho_{ij}(t_{m})$
for each $1\leq i,j\leq d$.

  2. (2)

Using the results, estimate the averaged coefficients $S_{ij}$ with

(43) $\sum_{m=1}^{M}\omega_{m}K_{\eta}(t_{m})\rho_{ij}(t_{m}).$

Numerical examples in two dimensions are described below in Section 4.2.

##  4 Numerical examples

###  4.1 Dimension $d=1$

We now present some one dimensional numerical examples of the proposed
averaging method for reducing the cell resonsance error. Since the error can
be computed either analytically or with quadrature formulas, it is relatively
easy to generate at large values of cell sizes $\eta$ relative to $\epsilon$
and hence detect convergence rates.

Figure 3 compares the cell resonance error (given by the difference between
(17) and the true harmonic average $\overline{a}$) with the averaged error
(20) for various kernels $K\in\mathbb{K}^{-p,q}$. We consider three
oscillatory functions $a^{\epsilon}(x)$ given by

$\displaystyle a^{\epsilon}_{1}(x)=$ $\displaystyle(1.1+\cos(2\pi
x/\epsilon))^{-1}$ $\displaystyle a^{\epsilon}_{2}(x)=$
$\displaystyle(2.2+\cos(2\pi x/\epsilon)+\cos(\sqrt{2}2\pi x/\epsilon))^{-1}$
(44) $\displaystyle a^{\epsilon}_{3}(x)=$
$\displaystyle(2+\operatorname{sign}(\cos(2\pi x/\epsilon)))^{-1}.$

The first two examples are smooth, while the third is discontinuous. In all
cases the true harmonic average, and, more generally, the incorrect average
$\rho^{\epsilon}(\eta)$ are easily computed analytically; note that the second
example is quasi-periodic, and hence the true harmonic average is given by
(17) as $\eta\to\infty$. The composite trapezoidal rule is used to compute the
weighted averages (20). We use a large number of quadrature points to ensure
the asymptotic decay rates are computed accurately; in particular when
$\eta=\epsilon$, $N_{\rm trap}=4096$ points are used, and the number of points
is increased proportionally as $\eta$ increases. All kernels
$K\in\mathbb{K}^{-p,q}$ that are used are of the form (13).

Figure 3:  A comparison of the absolute value of the one-dimensional cell
resonance error with the absolute value of the averaged error for various
averaging kernels $K\in\mathbb{K}^{-p,q}$ for the three oscillatory functions
$a^{\epsilon}$ given in (44).

In each case, the cell resonance error of course decays as $\epsilon/\eta$,
while the averaged errors decay at the rates predicted by Theorem 3.3 and
Theorem 3.4. Although the convergence rates in $\epsilon/\eta$ are higher for
averaging kernels with larger $p$ and $q$, the constant in front of the
asymptotic decay rate also increases. Empirically we find that $\left\lVert
K\right\rVert_{\infty}$ increases with $p$ and $q$, which impacts the estimate
(12) in Lemma 2.4. In contrast, the simple arithmetic average (using $K(x)=1$)
only decreases as $(\epsilon/\eta)^{2}$ as predicted by Theorem 3.4. However,
the asymptotic constant is relatively small, so it still results in a larger
decrease of the resonance error even at cell sizes $\eta\sim 50\epsilon$; _a
fortiori_ , it performs the best at “intermediate” values of $\eta$ (e.g.
$3\epsilon\lesssim\eta\lesssim 10\epsilon$) that are more practical in higher
dimensions.

###  4.2 Dimension $d=2$

We next present two dimensional numerical computations of the cell resonance
error as a function of domain size $\eta$ that suggest something analogous to
the one-dimensional decomposition from Lemma 3.1 indeed holds in higher
dimensions.

The cell problems (40) on $I_{\eta}=[-\eta/2,\eta/2]^{2}$ are solved with a
second order finite difference method implemented in C++ with the linear
algebra library Eigen [34]. $N=32$ grid points are used in both the $x_{1}$
and $x_{2}$ directions for the case $\eta=\epsilon$; as $\eta$ is increased,
$N$ is increased proportionally so that the resolution is approximately
constant for each problem. The estimates for the entries in the homogenized
tensor $\rho_{ij}(\eta)$ (given by (41)) are computed with the composite
trapezoidal rule, where the first order partial derivatives in the integrand
are approximated with a second order centered difference formula. The
estimates are computed for a sequence of domain sizes between $\eta=\epsilon$
and $\eta=16\epsilon$ with a spacing of $\Delta\eta=0.05\epsilon$.

We consider isotropic oscillatory tensors; that is, $a^{\epsilon}$ is given by
a (strictly positive) scalar-valued function times the identity matrix.
Consider first the $\epsilon$-periodic function

(45) $a^{\epsilon}(x_{1},x_{2})=\frac{2+1.5\sin(2\pi
x_{1}/\epsilon)}{2+1.5\sin(2\pi x_{2}/\epsilon)}+\frac{2+1.5\sin(2\pi
x_{2}/\epsilon)}{2+1.5\sin(2\pi x_{1}/\epsilon)},$

taken from Section 3.3 in [44] and visualized in Figure 4 (left). In this case
the true homogenized tensor $\overline{a}$ is diagonal but anisotropic. The
entries are approximated by solving the cell problem (40) for $\eta=\epsilon$
with $N=1024$ grid points in both the $x_{1}$ and $x_{2}$ directions, which
gives

$\overline{a}\simeq\begin{pmatrix}2.34348520086&0\\\
0&2.87329450077\end{pmatrix}.$

The resonance error $\rho_{ij}(\eta)-\overline{a}_{ij}$ for the two entries
$i=j=1$ and $i=j=2$ is also shown in Figure 4 (right); between the
$\epsilon/\eta$ decay envelopes the error is clearly oscillatory.

Figure 4:  Visualization of the periodic oscillatory function (45) (left) and
corresponding components of the cell resonance error as a function of domain
size (right).

Figure 5 (right) shows that the resonance error is also oscillatory for the
isotropic tensor given by the identity matrix times the quasi-periodic
function

(46)
$a^{\epsilon}(x_{1},x_{2})=\big{(}1.1+\cos(2\pi(x_{1}+\sqrt{2}x_{2})/\epsilon)\big{)}^{-1}.$

As in the the first case, the true homogenized tensor is anisotropic; in this
case however the coefficients $\overline{a}_{11}$ and $\overline{a}_{22}$ are
approximated by solving (40) at relatively large values of $\eta/\epsilon$. In
particular, we take coefficients to be the arithmetic average of the values
$\rho_{ij}(\eta)$ from $\eta=13\epsilon$ to $\eta=16\epsilon$ which gives

$\overline{a}\simeq\begin{pmatrix}1.75643523765&0\\\
0&1.34396605902\end{pmatrix}.$

Figure 5:  Visualization of the quasi-periodic oscillatory function (46)
(left) and corresponding components of the cell resonance error as a function
of domain size (right).

Given the oscillatory nature of the cell resonance error as a function of
domain size $\eta$, we now apply the numerical strategy outlined in Section
3.3 for its reduction to the cases (45) and (46). In both cases, the weighted
average (42) is approximated by the composite trapezoidal rule with intervals
of size $\Delta t=0.05\epsilon$.

Figure 6 shows the results for each of the two components of the resonance
error when $a^{\epsilon}$ is given by the $\epsilon$-periodic function (45).
We consider several types of kernels $K\in\mathbb{K}^{-p,q}$. As discussed in
Section 4.1 for the one dimensional results, kernels with $p\geq 1$ lead to
unsatisfactory results at the relatively low values of $\eta/\epsilon$
relevant for dimensions $d\geq 2$ owing to their large $\infty$-norms. In
contrast, the kernels $K(x)=1\in\mathbb{K}^{0,-1}([1,2])$ and

(47) $K(x)=Ce^{1/((x-1)(x-2))}\in\mathbb{K}^{0,\infty}([1,2])$

(where $C$ is chosen so that $K$ has unit mass on $[1,2]$) reduce the
resonance error by an order of magnitude or more at relatively low values of
$\eta/\epsilon$.

Figure 6:  A comparison of the absolute value of the resonance error with the
absolute value of the averaged error for various averaging kernels
$K\in\mathbb{K}^{-p,q}$ for the two-dimensional case (45).

Figure 7 shows the results for each of the two components of the resonance
error when $a^{\epsilon}$ is given by the quasi-periodic function (46). In
this case we only consider the kernels $K(x)=1$ and the exponential kernel
(47). As in the previous example, both reduce the resonance error by several
orders of magnitude.

Figure 7:  A comparison of the absolute value of the resonance error with the
absolute value of the averaged error for various averaging kernels
$K\in\mathbb{K}^{-p,q}$ for the two-dimensional case (46).

###  4.3 Computational cost and discussion

Assuming the estimate for the cell resonance error in Theorem 3.3 generalizes
to higher dimensions, the computational cost of the method proposed in Section
3.3 to reduce the error below some tolerance is asymptotically similar to the
cost of the hyperbolic microsolver proposed in [13]. Indeed, assume a linear
system with $N$ degrees of freedom can be solved with computational cost
proportional to $N^{\alpha}$; for example, $\alpha=2$ (respectively $2.5$) for
a sparse LU solver in dimension $d=2$ (resp. $d=3$). In an ideal scenario,
$\alpha=1$ could be achieved with a multigrid method, however, for microscale
domains with diameter $\eta$ sufficiently large relative to $\epsilon$, it is
not clear that the coarse grid solvers can resolve the oscillatory nature of
the solution. To generate the approximation (42) to the true homogenized
coefficients, the method proposed here consists of solving a sequence of
elliptic problems on domains with diameters that range from $\eta$ to $2\eta$.
Since (41) can oscillate with period $\epsilon$, in general
$\mathcal{O}\left(\eta/\epsilon\right)$ solves are required. Similarly, for
the hyperbolic solver a time step $\Delta t\sim\epsilon$ is required for a
total time of integration $T\sim\eta$. Hence, assuming that the total
computational cost to estimate the homogenized coefficients (including, e.g.
the use of quadrature formulas) is dominated by linear solves, both methods
have a total cost proportional to $(\eta/\epsilon)^{\alpha d+1}$. Of course,
to be fair it should again be emphasized that the authors in [13] rigorously
showed that the resonance error can be made to decay like
$(\epsilon/\eta)^{s}$, where $s$ can be arbitrarily large, while the same can
only be said for the current approach in dimension $d=1$.

Of all the existing techniques in the literature to reduce the cell resonance
error, in theory the modified elliptic problem recently proposed in [4] ought
to be the most computationally efficient, assuming the Arnoldi iterations used
to generate the forcing function $f$ (see Eq. (5)) converge sufficiently
rapidly; see section 4.3 in [4] for a discussion of this point. As with other
analyses found in the literature (see, for example, section 4.1 of [13]),
however, this assumes that the cell domain size $\eta$ is large enough that
the theoretical decay rates saturate, which may not be true until $\eta\sim
10\epsilon$ or larger. In practice a microsolver’s utility is determined by
its performance at relatively low values of $\eta$; depending on the
application, this could mean $\eta$ is several times the assumed local period,
or correlation length, $\epsilon$ of the heterogeneous media. A comparison of
the existing microsolvers in the literature at such “practical” values of
$\eta$ is beyond the scope of the present work, however such a study may be
valuable for streamlining research in numerical homogenization, especially if
done in collaboration with the engineering community [14].

One advantage of the present approach is that the microscale domain
calculations do not communicate with one another, i.e. the microscale solution
at a given domain size does not depend on any of the others. The efficiency of
the method hence can be greatly increased if the solutions are computed
independently in parallel.

Another feature unique to the present approach is that the form of the
microscale problem is unmodified from the standard elliptic problem from
homogenization theory; there is hence no need to develop approximations for
other operators. It is natural then to combine the approach with reduced basis
(RB) techniques for numerical homogenization, see for example [7, 8, 17, 43].
Such techniques are important for three dimensional problems, for which the
cost of solving microscale problems at every macroscale grid point where
homogenized coefficients are needed can be prohibitively expensive. They are
also important for optimal control problems constrained by multiscale PDEs.
Resolving the boundary resonance error in this context is an open problem, as
noted in [6].

Broadly speaking, RB methods precompute in an “offline” stage a low
dimensional set of basis functions (say of dimension $N$) by solving (40) at
different locations throughout the computational domain $\Omega$. Homogenized
coefficients needed by a macrosolver are then efficiently computed in an
“online” stage by solving linear systems of size $N\times N$ in the RB space.
The strategy described in Section 3.3 could be used to reduce the resonance
error for RB methods by first precomputing $M$ different basis sets, each
corresponding to a different domain size $t_{m}$, $1\leq m\leq M$. This of
course incurs a larger computational cost in the offline stage, although we
again note that this process is parallelizable. Once the different bases are
computed, however, the values {$\rho_{ij}(t_{m})\\}_{m=1}^{M}$ in (43) can be
easily obtained by solving $M$ low dimensional linear systems. We plan to
explore this possibility in future research.

##  5 Conclusions

A novel method for reducing the boundary resonance error in numerical
homogenization is proposed. Rather than modifying the form of the microscale
problem itself, the method is based on taking a weighted average of the
boundary error at a sequence of different microscale domain sizes.

Underlying the method is the observation that the boundary error itself is an
oscillatory function of domain size. In one dimensional and two dimensional
“tubular” domains the oscillatory nature is rigorously characterized.
Numerical evidence suggests the results also hold in more general settings,
however, more work is required to rigorously show that this is indeed the
case. The problem can be cast as a classical elliptic homogenization problem
with oscillatory boundary data, as considered in [29] and [26]; understanding
the potentially complicated boundary layers discussed therein is key for
further progress in this direction.

Based on the oscillatory nature of the boundary error, we propose solving
microscale problems for a sequence of domain sizes and then averaging the
results against kernels with vanishing “negative” moments designed to
accelerate the convergence of the resonance error to zero at infinitely large
microscale domains.

Owing to the rigorous analysis of the behavior of the resonance error, we show
for dimension $d=1$ that the error can be made to decay as
$(\epsilon/\eta)^{s}$ for $s$ arbitrarily large, which we confirm with
numerical experiments. However, the method is generally designed for higher
dimensions, and numerical examples in $d=2$ demonstrate its efficacy.

Because the form of the microscale problem is unchanged from classical
homogenization theory, the method does not require developing approximations
for other operators. In future work, the method could be combined with reduced
basis numerical homogenization techniques used for three dimensional problems
and for optimization problems with multiscale PDEs as constraints.

## Acknowledgments

The authors acknowledge support from the Oden Institute for Computational
Engineering and Sciences. We also thank Eric Hester for assistance with
Mathematica calculations to determine the averaging kernels used in some of
the numerical examples.

##  Appendix A Proof of results for averaging kernels

Proof of Lemma 2.2: By continuity of $g$, both the left and right-handed
limits exist for all $x$:

$\left|g^{[1]}(x\pm\Delta x)-g^{[1]}(x)\right|\to 0$

as $\Delta x\to 0$, so that $g^{[1]}$ is continuous. Furthermore,

$\displaystyle g^{[1]}(x+1)$
$\displaystyle=\int_{0}^{x+1}g(s)\,ds+c_{1}=\int_{0}^{x}g(s)\,ds+c_{1}+\int_{x}^{x+1}g(s)\,ds$
$\displaystyle=g^{[1]}(x)+\int_{0}^{1}g(s)\,ds=g^{[1]}(x)$

from the assumption that $g$ is mean zero. Lastly, note that continuous,
periodic functions are bounded. $\square$

The next proof is an adaptation of results from [12].

Proof of Lemma 2.4: Define $F^{[0]}(x)=f(x)-\left\langle f\right\rangle$, and
let $F^{[q+1]}$ be the $q+1$ primitive, as in Definition 2.1. Note

$\frac{d^{q+1}}{dx^{q+1}}F^{[q+1]}(x)=F^{[0]}(x),$

so that

(48)
$F^{[0]}(x\eta/\epsilon)=(\epsilon/\eta)^{q+1}\frac{d^{q+1}}{dx^{q+1}}F^{[q+1]}(x\eta/\epsilon).$

Furthermore, note that $\left\lVert F^{[q+1]}\right\rVert_{\infty}\leq
2^{q+1}\left\lVert f\right\rVert_{\infty}$. Then after a change of variables,

$\int_{\eta}^{2\eta}K_{\eta}(x)x^{-r}f\left(\frac{x}{\epsilon}\right)dx=\underbrace{\eta^{-r}\int_{1}^{2}K(u)u^{-r}F^{[0]}(u\eta/\epsilon)du}_{(\textrm{I})}+\underbrace{\left\langle
f\right\rangle\eta^{-r}\int_{1}^{2}K(u)u^{-r}du,}_{(\textrm{II})}$

where $F^{[0]}$ is mean-zero. Since $K\in\mathbb{K}^{-p,q}$, $(\textrm{II})$
vanishes if $r\leq p$; otherwise,

$\left|\left\langle
f\right\rangle\eta^{-r}\int_{1}^{2}K(u)u^{-r}du\right|\leq\left|\left\langle
f\right\rangle\right|\left\lVert K\right\rVert_{\infty}\,\eta^{-r}.$

After using (48) and integrating by parts, $(\textrm{I})$ becomes

$\displaystyle(\textrm{I})$
$\displaystyle=(\epsilon/\eta)^{q+1}\eta^{-r}\int_{1}^{2}K(u)u^{-r}\frac{d^{q+1}}{du^{q+1}}F^{[q+1]}(u\eta/\epsilon)du$
$\displaystyle=(-1)^{q+1}(\epsilon/\eta)^{q+1}\eta^{-r}\int_{1}^{2}\frac{d^{q+1}}{du^{q+1}}\left(K(u)u^{-r}\right)F^{[q+1]}(u\eta/\epsilon)du$

Since $u^{-r}\in C^{\infty}([1,2])$ and by assumption $K^{(q+1)}\in
BV([1,2])$,

$g(u):=\frac{d^{q+1}}{du^{q+1}}\left(K(u)u^{-r}\right)\in BV([1,2])$

as well. Changing variables again and breaking the resulting integral in three
pieces gives

$\displaystyle(\textrm{I})$
$\displaystyle=(-1)^{q+1}(\epsilon/\eta)^{q+2}\eta^{-r}\int_{\eta/\epsilon}^{2\eta/\epsilon}g(t\epsilon/\eta)F^{[q+1]}(t)\,dt$
$\displaystyle=(-1)^{q+1}(\epsilon/\eta)^{q+2}\eta^{-r}\Big{(}\int_{\eta/\epsilon}^{\lceil\eta/\epsilon\rceil}g(t\epsilon/\eta)F^{[q+1]}(t)\,dt+\int_{\lceil\eta/\epsilon\rceil}^{\lfloor
2\eta/\epsilon\rfloor}g(t\epsilon/\eta)F^{[q+1]}(t)\,dt$
$\displaystyle\,\,\,\,\qquad\qquad\qquad\qquad\qquad+\int_{\lfloor
2\eta/\epsilon\rfloor}^{2\eta/\epsilon}g(t\epsilon/\eta)F^{[q+1]}(t)\,dt\Big{)}$

The first integral in parentheses can be bounded as

$\Big{|}\int_{\eta/\epsilon}^{\lceil\eta/\epsilon\rceil}g(t\epsilon/\eta)F^{[q+1]}(t)\,dt\Big{|}\leq\left\lVert
g\right\rVert_{\infty}\left\lVert F^{[q+1]}\right\rVert_{\infty}\leq
2^{q+1}\left\lVert f\right\rVert_{\infty}\,V_{1}^{2}(g)$

where $V_{1}^{2}(g)$ denotes the total variation of $g$ on $[1,2]$ and depends
only on $K$, its derivatives, and $q$. A similar bound exists for the third
integral. Next, let $N=\lfloor
2\eta/\epsilon\rfloor-\lceil\eta/\epsilon\rceil$. By construction, $F^{[q+1]}$
is a mean-zero, 1-periodic function, so that

$\int_{\lceil\eta/\epsilon\rceil}^{\lfloor
2\eta/\epsilon\rfloor}F^{[q+1]}(t)\,dt=0.$

The second integral is then bounded by

$\Big{|}\sum_{j=1}^{N-1}\int_{\lceil\eta/\epsilon\rceil
j}^{\lceil\eta/\epsilon\rceil(j+1)}\big{(}g(t\epsilon/\eta)-g(\lceil\eta/\epsilon\rceil
j\epsilon/\eta)\big{)}F^{[q+1]}(t)\,dt\Big{|}\leq 2^{q+1}\left\lVert
f\right\rVert_{\infty}\,V_{1}^{2}(g).$

Putting it all together, we’ve shown that

$\Big{|}\int_{\eta}^{2\eta}K_{\eta}(x)x^{-r}f\left(\frac{x}{\epsilon}\right)dx\Big{|}\leq
3\eta^{-r}\left(\frac{\epsilon}{\eta}\right)^{q+2}2^{q+1}\left\lVert
f\right\rVert_{\infty}V_{1}^{2}(g)+\left|\left\langle
f\right\rangle\right|\left\lVert K\right\rVert_{\infty}\,\eta^{-r}$

for $p<r$ and

$\Big{|}\int_{\eta}^{2\eta}K_{\eta}(x)x^{-r}f\left(\frac{x}{\epsilon}\right)dx\Big{|}\leq
3\eta^{-r}\left(\frac{\epsilon}{\eta}\right)^{q+2}2^{q+1}\left\lVert
f\right\rVert_{\infty}V_{1}^{2}(g)$

for $p\geq r$, as desired. $\square$

##  Appendix B Proof of Theorem 3.4

Proof: Following the proof of Theorem 3.3, we use the expansion (19) from
Lemma 3.1 and then take the arithmetic average on the interval $[\eta,2\eta]$.
Since the expansion converges absolutely for $\eta/\epsilon>\overline{a}B$,
this means

(49)
$\frac{1}{\eta}\int_{\eta}^{2\eta}\rho^{\epsilon}(x)\,dx-\overline{a}=\overline{a}\sum_{k=1}^{\infty}\frac{1}{\eta}\int_{\eta}^{2\eta}\left(-\frac{\epsilon}{x}q(x/\epsilon)\right)^{k}dx$

where

$q(x/\epsilon)=\overline{a}\int_{0}^{x/\epsilon}(b(s)-\left\langle
b\right\rangle)\,ds.$

Considering first the $k=1$ term in the sum and integrating by parts gives

$\displaystyle-\frac{1}{\eta}\int_{\eta}^{2\eta}\frac{\epsilon}{x}q(x/\epsilon)\,dx$
$\displaystyle=-\frac{1}{\delta}\int_{\delta}^{2\delta}\frac{q(u)}{u}\,du$
(50)
$\displaystyle=\frac{1}{\delta}\int_{\delta}^{2\delta}\frac{Q(u)}{u^{2}}\,du-\frac{1}{\delta}\left(\frac{1}{2\delta}Q(2\delta)-\frac{1}{\delta}Q(\delta)\right),$

where $\delta:=\eta/\epsilon$ and $Q$ is the primitive of $q$ as defined by
Definition 2.1. Since $b(u)-\left\langle b\right\rangle$ is a mean-zero
periodic function, Lemma 2.2 guarantees that $q(u)$ is also a periodic
function. Moreover, since $b(s)=1/a(s)$ is assumed to be an even function,
$q(u)$ is an odd function, and hence

$\int_{-1/2}^{1/2}q(u)\,du=0.$

Because $q$ is a mean-zero, one-periodic function, Lemma 2.2 implies that $Q$
is also periodic, and hence bounded independently of $\delta$. Ergo, by the
triangle inequality (50) implies

(51)
$\left|\frac{1}{\eta}\int_{\eta}^{2\eta}\frac{\epsilon}{x}q(x/\epsilon)\,dx\right|\leq
2\left\lVert Q\right\rVert_{\infty}\left(\frac{\epsilon}{\eta}\right)^{2}$

It then remains to consider the $k\geq 2$ terms of the sum in (49). As in (25)
and (26),

$\sup_{x\in\mathbb{R}}\left|q(x/\epsilon)\right|\leq\overline{a}B$

and hence

$\sup_{x\in\mathbb{R}}\left|(q(x/\epsilon))^{k}\right|\leq\overline{a}^{k}B^{k}.$

We then have

$\overline{a}\left|\sum_{k=2}^{\infty}\frac{1}{\eta}\int_{\eta}^{2\eta}\left(-\frac{\epsilon}{x}q(x/\epsilon)\right)^{k}dx\right|\leq\overline{a}\sum_{k=2}^{\infty}(\overline{a}B)^{k}\left(\frac{\epsilon}{\eta}\right)^{k}=\overline{a}\,(\overline{a}B)^{2}\left(\frac{\epsilon}{\eta}\right)^{2}\varphi\big{(}\overline{a}B\frac{\epsilon}{\eta}\big{)}.$

Along with (51), this gives the desired result.

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# Is a Compact Group with All Dense Subgroups Separable Metrizable?

Dekui Peng<EMAIL_ADDRESS>Institute of Mathematics, Nanjing Normal
University, Nanjing 210046, China

###### Abstract.

A compact group with all dense subspaces separable is metrizable. Inspired by
this, we conjecture that a compact group with all dense subgroups separable is
metrizable. Positive answers are given here for two elementary cases, say,
when the compact group is additionally assumed to be abelian or connected.
However, a locally compact abelian group, even when it has an open compact
subgroup, with all dense subgroups separable, may not be metrizable. At the
end of this note, it is shown that a locally compact group with all subgroups
separable is metrizable. Our arguments are formalized in a more general form,
namely, not restricted to the countable case.

###### Key words and phrases:

compact groups; separable; metrizable

###### 2010 Mathematics Subject Classification:

Primary 22C05; Secondary 22D05, 54A25

This work is supported by NSFC grant NO. 12271258.

##  1\. Introduction

The density $d(X)$ of a topological space $X$ is the minimum cardinal $\tau$
such that $X$ admits a dense subset of size $\tau$. We denote by $w(X)$ the
weight of $X$, then $d(X)\leq w(X)$. A topological space with countable
density is called separable.

As is known, unlike the metrizable case, in a general separable topological
space a dense subspace need not be separable. So it is an interesting topic to
study the densities of dense subspaces of a (separable) topological space in
General Topology. In a very recent paper [8] by I. Juhász, J. van Mill, L.
Soukup, and Z. Szentmiklóssy, the double density spectrum $dd(X)$ of a
topological space $X$ is defined; it is the set of densities of dense
subspaces of $X$, i.e.,

$dd(X)=\\{d(Y)|\mbox{$Y$~{}is~{}a~{}dense~{}subspace~{}of~{}$X$}\\}.$

Some interesting results about the double density spectra for compact spaces
were given in their Section 3. Juhász and van Mill’s further study on double
density spectra [7] shows that $dd(G)$ is the interval $[d(G),w(G)]$ for any
locally compact group $G$. This particular implies that $w(G)=d(G)$ if
$dd(G)=\\{d(G)\\}$ for locally compact groups $G$. If $G$ is separable, the
result becomes:

######  Proposition 1.1.

If $G$ is a locally compact group with all dense subspaces separable, then $G$
is metrizable.

In the study of topological groups, we usually care about subgroups more than
subspaces. So it is natural to ask whether one may replace “subspaces” by
“subgroups”. We shall give a counterexample to this question later.
Nevertheless, I guess the following is true.

######  Conjecture 1.2.

A compact group $G$ with all dense subgroups separable is metrizable.

Positive answers to the conjecture will be given in the abelian case and the
connected case. Indeed, we prove these cases for any cardinals:

######  Theorem A.

Let $\tau$ be an infinite cardinal and $G$ a compact group which is abelian or
connected such that every dense subgroup of $G$ has density $\leq\tau$. Then
$w(G)\leq\tau$.

The abelian case of this conjecture is also considered by F. Lin, Q. Wu and C.
Liu [9], who provided partial answers. For abelian case, our proof avoids the
results of them, and the main tool is the dual theorem for precompact abelian
groups discovered by W. Comfort and K. Ross [1] and further developed by many
people, including the author [10]. For the proof of connected case, the
structure theorems in the book [3] of K. Hofmann and S. Morris are used.

Recall that a topological space is called hereditarily separable if all
subspaces of it are separable. It is a known result of R. Engelking that every
non-metrizable compact group of weight $\tau$ contains the Cantor cube
$D^{\tau}$ as a subspace, where $D$ is the two-point discrete space. Since
$\tau>\omega$, the space $D^{\tau}$ is not hereditarily separable. This
implies that a hereditarily separable compact group is metrizable. We shall
generalize the result to the following.

######  Theorem B.

Let $\tau$ be an infinite cardinal and $G$ a locally compact group such that
every subgroup of $G$ has density $\leq\tau$. Then $w(G)\leq\tau$.

###  1.1. Notation and Terminology

All topological spaces are assumed to be Hausdorff, unless otherwise stated.
Abelian groups are written additively, including the torus $\mathbb{T}$, so
the neutral element of an abelian group is denoted by $0$. For $n$ an integer
and $G$ an abelian group, $nG$ is the subgroup $\\{ng:g\in G\\}$. Let $X$ be a
subset of a group $G$, the subgroup of $G$ generated by $X$ is denoted by
$\langle X\rangle$. If $X=\\{g\\}$, we shall write simply $\langle g\rangle$
instead of $\langle\\{g\\}\rangle$. We denote by $\mathbb{Z}$ the additive
group of integers, by $\mathbb{Q}$ the rationals and by $\mathbb{R}$ the
reals. For any positive integer $n$, $\mathbb{Z}(n)$ is the cyclic group with
$n$ elements $\mathbb{Z}(p^{\infty})$ is the Prüfer group for a prime $p$.

Let $\\{G_{\alpha}:\alpha<\tau\\}$ be a family of topological groups and the
identity of each $G_{\alpha}$ is denoted by $1$. For any element
$g=(g_{\alpha})_{\alpha<\tau}\in\prod_{\alpha<\tau}G_{\alpha}$, the support of
$g$ is the subset $supp(g)=\\{\alpha<\tau:g_{\alpha}\neq 1\\}$ of $\tau$. The
set of all elements in $\prod_{\alpha<\tau}G_{\alpha}$ with finite support is
a (dense) subgroup and called the restricted product of
$\\{G_{\alpha}:\alpha<\tau\\}$, which is also known as the direct sum (most
usually in abelian case). For a group $G$ and a cardinal $\tau$, $G^{(\tau)}$
is the restricted product of $\tau$ many copies of $G$.

A topological group is called precompact if it is a dense subgroup of a
compact group. Now consider $G$ as an abstract abelian group. Then the
Comfort-Ross Duality [1] yields that every precompact group topology $\sigma$
on $G$ is generated by a dense subgroup $D$ of $\widehat{G}$, i.e., $\sigma$
is the weakest one making all characters $\chi(:G\to\mathbb{T})\in D$
continuous. In this case, $D$ is called the Comfort-Ross dual group of
$(G,\sigma)$ and denoted by $(G,\sigma)^{*}$. If $f:G\to H$ is a continuous
homomorphism of precompact abelian groups, then there exists a continuous
homomorphism $f^{*}:H^{*}\to G^{*}$ in a natural way (see [10]). The following
elementary facts of Comfort-Ross Duality, which can be found in [10], will be
used in this note.

######  Fact 1.3.

Let $G$ and $H$ be precompact abelian groups, then

  * (a)

$G\cong(G^{*})^{*}$;

  * (b)

$w(G^{*})=|G|$; and

  * (c)

$f:G\to H$ is a continuous isomorphism if and only if $f^{*}:H^{*}\to G^{*}$
is a topological group embedding with dense image.

Throughout of the paper, $\omega$ is always the first infinite (countable)
cardinal. For a set $X$, $|X|$ is the cardinality of $X$, $d(X)$ is the
density of $X$ and $w(X)$ is the weight. If $G$ is a topological group and $N$
a closed subgroup, then it is well-known and easily verified that
$w(G)=w(N)w(G/N)$, where $G/N$ is the space of left coset space.

For an infinite cardinal number $\tau$, a subset of a topological space $X$ is
called a $G_{\tau}$ set if it is the intersection of (at most) $\tau$ many
open subsets of $X$. In frequently used terminology, $G_{\omega}$ sets are
known as $G_{\delta}$ sets.

######  Lemma 1.4.

Let $G$ be a precompact abelian group and $\tau$ an infinite cardinal. Then
$d(G)\leq\tau$ if and only if $\\{0\\}$ is a $G_{\tau}$ set in $G^{*}$.

###### Proof.

First assume that $d(G)\leq\tau$. Then $G$ admits a dense subgroup $H$ of size
$\leq\tau$, where $\tau\geq\omega$, with $j:H\to G$ the embedding. Then Fact
1.3 (b) yields that $w(H^{*})\leq\tau$; so $\\{0\\}$ is a $G_{\tau}$ set in
$H^{*}$. Moreover, by Fact 1.3 (c), $j^{*}:G^{*}\to H^{*}$ is a continuous
isomorphism. Hence, $\\{0\\}$ is also a $G_{\tau}$ set in $G^{*}$.

Conversely, if the singleton $\\{0\\}$ is a $G_{\tau}$ set in $G^{*}$, then
there exists a family $\\{U_{\alpha}:\alpha<\tau\\}$ of open sets in $G^{*}$
such that $\bigcap_{\alpha<\tau}U_{\alpha}=\\{0\\}$. For each $\alpha<\tau$,
one may take a continuous homomorphism $\chi_{\alpha}:G^{*}\to\mathbb{T}$ such
that $\ker\chi_{\alpha}\subseteq U_{\alpha}$. Then the diagonal product
$\chi:=\Delta_{\alpha}\chi_{\alpha}:G^{*}\to\mathbb{T}^{\tau}$ is injective.
In particular, $w(\chi(G^{*}))\leq\tau$. Let $\varphi$ be the restriction of
$\chi$ to its image, i.e., $\varphi$ is the morphism $G^{*}\to\chi(G^{*})$
sending $x\to\chi(x)$, for all $x\in G^{*}$. Then
$\varphi^{*}:(\chi(G^{*}))^{*}\to(G^{*})^{*}$ is an embedding by Fact 1.3 (c).
The other two items in Fact 1.3 then implies that $G$ has a dense subgroup of
$\leq\tau$, i.e., $d(G)\leq\tau$. ∎

##  2\. Proof of Theorem A

###  2.1. Abelian Case

We first begin with the following lemma.

######  Lemma 2.1.

Let $\tau$ be an infinite cardinal. Then every abelian group $G$ with
$|G|>\tau$ admits a precompact group topology under which the singleton
$\\{0\\}$ is not a $G_{\tau}$ set.

###### Proof.

Let $G$ be an abelian group with $|G|=\kappa>\tau$ and $D$ the divisible hull
of $G$. Then $D$ has cardinality $\kappa$ and $G$ is an essential subgroup of
$D$, i.e., $G\cap N\neq\\{0\\}$ for each $\\{0\\}\neq N\leq D$. By the
structure theorem of divisible abelian groups, $D$ splits into the direct sum
of the groups $\\{D_{\alpha}:\alpha<\kappa\\}$, where each $D_{\alpha}$ is
isomorphic to either $\mathbb{Q}$ or $\mathbb{Z}(p^{\infty})$ for some prime
$p$. In both cases $D_{\alpha}$ embeds into the circle $\mathbb{T}$ as a dense
subgroup. Thus, one embeds $\Pi:=\prod_{\alpha<\kappa}D_{\alpha}$ into
$\mathbb{T}^{\kappa}$. Hence $D=\bigoplus_{\alpha<\kappa}D_{\alpha}$, which is
isomorphic to the restricted product of $\\{D_{\alpha}:\alpha<\tau\\}$,
obtains a precompact group topology.

We first claim that under this topology $\\{0\\}$ is not $G_{\tau}$ in $D$.
Suppose for the contrary that $\\{0\\}$ is a $G_{\tau}$ set in $D$. Then there
exists a family $\\{U_{\alpha}:\alpha<\tau\\}$ of open subsets of $\Pi$ such
that $\bigcap_{\alpha<\tau}U_{\alpha}\cap D=\\{0\\}$. For each $\alpha<\tau$,
there exists a finite subset $E_{\tau}$ of $\kappa$ such that
$\prod_{\alpha\in\kappa\setminus E_{n}}D_{\alpha}\subseteq U_{\tau}$. Let
$E=\bigcup_{\alpha<\tau}E_{n}$ is a subset of $\kappa$ of cardinality $\tau$
and $\prod_{\alpha\in\kappa\setminus
E}D_{\alpha}\subseteq\bigcap_{\alpha<\tau}U_{\alpha}$. Take
$\beta\in\kappa\setminus E$. Then
$D_{\beta}\subseteq\bigcap_{\alpha<\tau}U_{\alpha}\cap D$. This is a
contradiction.

Now let $P$ be any $G_{\tau}$ set of $D$ containing $0$. Without loss of
generality, one may assume that $P$ is a subgroup111One may assume
$P=\bigcap_{\alpha\in\tau}U_{\alpha}$ such that $U_{\alpha}$ is open in $D$
and $U_{\alpha+1}U_{\alpha+1}\subset U_{\alpha}$ for all $\alpha<\tau$. Then
for every limit ordinal $\beta<\tau$, $\bigcap_{n\in\omega}U_{\beta+n}U_{n}$
is a subgroup; therefore, $P$ is a intersection of subgroups.. By the above
assertion, $P\neq\\{0\\}$. The essentiality of $G$ in $D$ then yields that
$P\cap G\neq\\{0\\}$. Thus, $\\{0\\}$ cannot be a $G_{\tau}$ set in $G$. ∎

Now we are ready to give the proof.

###### Proof of Theorem A: Abelian Case.

Let $G$ be a compact abelian group with $w(G)=\kappa>\tau$ and $H=\widehat{G}$
the (discrete) Pontryagin dual group of $G$. Then $H$ admits a precompact
group topology under which the singleton $\\{0\\}$ is not a $G_{\tau}$ set,
according to Lemma 2.1. Then the Comfort-Ross dual group $K$ of $H$ ($H$ is
given the announced topology) is a dense subgroup of $G$, which is not of
density $\leq\tau$ by Lemma 1.4. ∎

###  2.2. Two Non-compact Counterexamples

One may wonder can Theorem A (abelian case) can be extended to wider classes
of topological groups. Two natural generalizations compactness in topological
groups are precompactness and local compactness. However, Theorem A (abelian
case) can be extended to neither of these cases.

######  Example 2.2.

There are non-metrizable locally compact abelian groups and non-metrizable
precompact abelian groups whose all dense subgroups are separable.

  * (1)

Recall that a topological group is called topologically simple if it contains
exactly two closed normal subgroups. In [5] and [10], it is shown that non-
metrizable, topologically simple, precompact abelian groups exist. Let $G$ be
such a group and $H$ a dense subgroup. Take $0\neq h\in H$, then $\langle
h\rangle$ is a non-trivial subgroup of $G$. By topological simpleness of $G$,
$\langle h\rangle$ is dense in $G$, so also in $H$. This particularly shows
that $H$ is separable.

  * (2)

Let $\tau$ be an uncountable cardinal such that $\tau\leq 2^{\omega}$ and
$K=\mathbb{Z}(2)^{\tau}$ with the usual product topology. By the celebrated
Hewwit-Marczewski-Pondiczery Theorem [2, Theorem 2.3.15], $K$ is separable.
Let $D$ be a countable dense subgroup of $K$. Pure algebraically, $D$ is a
direct summand of $K$, since it is a subspace of the linear space $K$ over
$\mathbb{Z}(2)$. Now write $K=D\oplus E$ for some subgroup (or linear
subspace) $E$ of $K$. It is evident that $E$ is isomorphic to $K$, of course,
only in the algebraic sense. Let $G=\mathbb{Z}(4)^{(\omega)}\oplus E$. Then
$K=D\oplus E$ embeds into $G$ as a subgroup in the natural way: one consider
$D$ as the socle of $Z(4)^{(\omega)}$, i.e., $D=\\{2x:x\in
Z(4)^{(\omega)}\\}$, and identify the $E$ in $K$ with $E$ in $G$. So

$2G=2(Z(4)^{(\omega)})+2E=D+\\{0\\}=D.$

Let $G$ have the group topology making $K$ an open subgroup. Then $G$ is
evidently locally compact and non-metrizable. It remains to check that every
dense subgroup $H$ of $G$ is separable.

Since $K$ is open in $G$ and $H$ is dense, $G=K+H$. Then

$D=2G=2K+2H=\\{0\\}+2H=2H\leq H.$

Take a countable subset $P$ of $H$ contains a representative of every coset of
$K$ in $G$. Then $H^{\prime}=\langle P\rangle+D$ is a countable subgroup of
$H$. By the choice of $P$, for each $g\in G$, there exists $p_{g}\in P$ such
that $p_{g}+K=g+K$. So $H^{\prime}\cap(g+K)$ contains $p_{g}+D$, which is
dense in $g+K$. In summary, $H^{\prime}$ contains a dense subset for each
coset of $K$ in $G$, so $H^{\prime}$ is dense in $G$.

###  2.3. Connected Case

Let us recall the following Theorem known as Levi-Mal’cev Structure Theorem
(for connected compact groups). For any topological group $G$, $Z(G)$ and
$G^{\prime}$ will be the center and (abstract) commutator subgroup,
respectively. And $Z_{0}(G)$ is the identity component of $Z(G)$.

######  Theorem 2.3.

[3, Theorem 9.24] Let $G$ be a connected compact group and
$\Delta=Z_{0}(G)\cap G^{\prime}$. Then $G=Z_{0}(G)G^{\prime}$ and there exist
a family $\\{S_{j}|j\in J\\}$ of simple simply connected compact Lie groups
and a closed totally disconnected central subgroup $C$ of $G$ such that

  * (i)

$G^{\prime}$ is a quotient group of $\prod_{j\in J}S_{j}$ (with the kernel
contained in $\prod_{j\in J}Z(S_{j})$); and

  * (ii)

$G/C\cong(Z_{0}(G)/\Delta)\times\prod_{j\in J}(S_{j}/Z(S_{j}))$.

The following two lemmas are “folklore”. While, we prefer to give the proofs
for the readers’ convenience.

######  Lemma 2.4.

Let $\tau$ be an infinite cardinal. If $G$ is a topological group with all
dense subgroups are of density $\leq\tau$, then also is every continuous
homomorphism image of $G$.

###### Proof.

Let $H$ be a continuous homomorphism image of $G$. Then the preimage of $D$ in
$G$ is a dense subgroup; so admits a dense subset of cardinality $\leq\tau$.
As a continuous image, $D$ also has a such a dense subset.∎

######  Lemma 2.5.

Let $\kappa$ be an infinite cardinal and $\\{H_{\alpha}:\alpha<\kappa\\}$ a
family of non-trivial topological groups. Then the restricted product $H$ of
$\\{H_{\alpha}:\alpha<\kappa\\}$ has density $\geq\kappa$.

###### Proof.

Let $P=\prod_{\alpha<\kappa}H_{\alpha}$. Then $H$ is a dense subgroup of $P$.
Let $C$ be a subset of $H$ of size $<\kappa$. Then the cardinality of
$supp(C):=\bigcup_{g\in C}supp(g)$ is strictly less than $\kappa$ as well,
since each $supp(g)$ is finite. Let $\overline{C}$ be the closure of $C$ in
$P$, then every $h\in\overline{C}$ satisfies $supp(h)\subseteq supp(C)$. So
$\overline{C}$ cannot be the whole $P$, i.e., $C$ is not dense in $P$ or $H$.
∎

###### Proof of Theorem A: Connected Case.

Let us use the notations as in Theorem 2.3. By Lemma 2.4 and (ii) in Theorem
2.3, all dense subgroups of $Z_{0}(G)/\Delta$ and all dense subgroups of
$\prod_{j\in J}(S_{j}/Z(S_{j}))$ are of density $\leq\tau$. Using Lemma 2.5,
one obtains immediately that $|J|\leq\tau$. This implies that

(2.1) $w(G^{\prime})\leq\tau$

as $G^{\prime}$ is a quotient group of $\prod_{j\in J}S_{j}$. On the other
hand, $Z_{0}(G)/\Delta$ is abelian. So our assertion for abelian case implies
that

(2.2) $w(Z_{0}(G)/\Delta)\leq\tau.$

As a subgroup of $G^{\prime}$, $\Delta$ satisfies

(2.3) $w(\Delta)\leq\tau$

by (2.1). Combining the inequalities (2.2) and (2.3), one obtain

(2.4) $w(Z_{0}(G))\leq\tau.$

For the last step, remind that $G=Z_{0}(G)G^{\prime}$. So $G$ is a continuous
image of $Z_{0}(G)\times G^{\prime}$. Then (2.1) together with (2.4) will
imply that $w(G)\leq\tau$. ∎

##  3\. Proof of Theorem B

We need the so-called Countable Layer Theorem for compact groups discovered by
K. Hofmann and S. Morris as following:

######  Theorem 3.1.

[4] Let $G$ be a compact group. Then there exists a chain of closed
characteristic subgroups (may stops after finite steps)

$G=G_{0}\geq G_{1}\geq G_{2}\geq...\geq\bigcap_{n\in\omega}G_{n}=Z_{0}(G)$

of $G$ such that the factor $G_{n}/G_{n+1}$ is a product of simple compact
groups222A simple compact group is either a simple connected compact Lie group
with trivial center or a finite simple group. for each $n\in\omega$.

###### Proof of Theorem B.

We first deal with compact groups $G$. For every natural number, $G_{n}$ is
the subgroup of $G$ announced in Theorem 3.1. By our assumption, for each
$n\in\omega$, all (dense) subgroups of $G_{n}$ are of density $\leq\tau$. Then
Lemma 2.4 yields particular that all dense subgroups of $G_{n}/G_{n+1}$ have
density not greater than $\tau$ as well. Now apply Lemma 2.5, one obtains that
$G_{n}/G_{n+1}$ is the product of at most $\tau$ many simple compact groups.
It is evident that simple compact groups are metrizable, so

$w(G_{n}/G_{n+1})\leq\tau.$

By induction, one obtain that $w(G/G_{n})\leq\tau$ for all $n\in\omega$. Note
that $G/Z_{0}(G)$ is the inverse limit of the inverse system
$\\{G/G_{n};\varphi^{j}_{i}\\}$ indexed by $\omega$, where for each pair
$i<j$, $\varphi^{j}_{i}:G/G_{j}\to G/G_{i}$ is the canonical quotient mapping.
So $G/Z_{0}(G)$ embeds into the product of countably many groups with weight
$\leq\tau$; thus

$w(G/Z_{0}(G))\leq\tau.$

It remains to prove the following

$w(Z_{0}(G))\leq\tau.$

While, this follows from the abelian case of Theorem A immediately.

Let us be back to the general case that $G$ is only locally compact. Denote by
$G_{0}$ the identity component of $G$. Then $G/G_{0}$ is totally disconnected.
The well-known van Dantzig Theorem yields that $G/G_{0}$ has an open compact
subgroup $U$. Let $V$ be the preimage of $U$ in $G$, then $V$ is an open
subgroup of $G$ containing $G_{0}$ and $V/G_{0}$ is compact. The solution of
Hilbert’s fifth problem then yields that $V$ contains a compact normal
subgroup $N$ such that $V/N$ is a Lie group. In particular,

$w(V/N)=\omega\leq\tau.$

And by the former argument for compact groups, one knows that $w(N)\leq\tau$.
So $w(V)\leq\tau$. It remains to note that $|G/V|\leq\tau$ because $V$ is open
in $G$ and $d(G)\leq\tau$. ∎

We have seen that Theorem A cannot be generalized to locally compact abelian
groups. So one may wonder that whether or not we can generalize this theorem
in the other direction.

######  Question 3.2.

Let $\tau$ be an infinite cardinal and $G$ a connected locally compact group
such that every dense subgroup of $G$ is of density $\leq\tau$. Is
$w(G)\leq\tau$?

Let me give some discussion on this question. Again, by the solution of
Hilbert’s fifth problem, $G$ has an compact normal subgroup $N$ such that
$G/N$ is a Lie group. Evidently, one may assume that $N$ is connected as well.
Let $H$ be the centralizer of $N$ in $G$. By Theorem 2 of the celebrated paper
[6] of K. Iwasawa, $G=HN$.

Remind that $N=Z(N)N^{\prime}$, so we have

(3.1) $HN^{\prime}=(HZ(N))N^{\prime}=HN=G,$

and

$Z(N)\cap N^{\prime}=Z(N^{\prime}).$

Let $K=Z(N^{\prime})$. Since $H\cap N^{\prime}=(H\cap N)\cap
N^{\prime}=Z(N)\cap N^{\prime}=K$, by (3.1) we have

(3.2) $G/K\cong H/K\times N^{\prime}/K.$

As in Theorem 2.3 (i), there exists a continuous surjective homomorphism
$\pi:\prod_{j\in J}S_{j}\to N^{\prime}$ such that $\ker\pi\leq\prod_{j\in
J}Z(S_{j})$. It is known that a continuous surjective homomorphism between
connected compact groups sends the center onto the center, see [3, Theorem
9.28]. So $K=\pi(\prod_{j\in J}Z(S_{j}))$ and therefore,
$N^{\prime}/K\cong\prod_{j\in J}(S_{j}/Z(S_{j}))$. In summary, we obtain that
$\prod_{j\in J}(S_{j}/Z(S_{j}))$ is a continuous homomorphism image of $G$. By
Lemmas 2.4 and 2.5, we have $|J|\leq\tau$. So the weight of $N^{\prime}$ is at
most $\tau$.

Now Question 3.2 is reduced into the special case:

######  Question 3.3.

Let $\tau$ be an infinite cardinal and $G$ a connected locally compact group
such that every dense subgroup of $G$ is of density $\leq\tau$. If $G$ has an
compact central subgroup $C$ such that $G/C$ is a Lie group, is
$w(G)\leq\tau$?

Indeed, since we have proved $w(N^{\prime})\leq\tau$ and hence $w(K)\leq\tau$,
it remains to show that

(3.3) $w(H/K)\leq\tau,$

according to the equality (3.2). As a continuous homomorphism image of $G$,
$H/K$ has all its dense subgroups of density $\leq\tau$. Moreover, $Z(N)/K$ is
a central subgroup of $H/K$ and

$(H/K)/(Z(N)/K)\cong H/Z(N)\cong G/N$

is a Lie group. Hence, if one proved Question 3.3, then (3.3) would hold.

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