Estimates of the modeling error generated by homogenization of an elliptic boundary value problem

Sergey Repin; Tatiana Samrowski; Stefan Sauter

doi:10.1515/jnma-2014-1002

Publicly Available Published by De Gruyter March 14, 2016

Estimates of the modeling error generated by homogenization of an elliptic boundary value problem

Sergey Repin , Tatiana Samrowski and Stefan Sauter

From the journal Journal of Numerical Mathematics

https://doi.org/10.1515/jnma-2014-1002

Abstract

In this paper, we derive a posteriori bounds of the difference between the exact solution of an elliptic boundary value problem with periodic coefficients and an abridged model, which follows from the homogenization theory. The difference is measured in terms of the energy norm of the basic problem and also in the combined primal–dual norm. Using the technique of functional type a posteriori error estimates, we obtain two-sided bounds of the modelling error, which depends only on known data and the solution of the homogenized problem. It is proved that the majorant with properly chosen arguments possesses the same convergence rate, which was established for the true error. Numerical tests confirm the efficiency of the estimates.

Keywords: periodic structures; homogenization; elliptic boundary value problems; a posteriori error estimates; modeling error

MSC 2010: 35J15; 35B27; 65N15

1 Introduction

Boundary value problems with periodic structures arise in various applications. Homogenization theory is the major tool used to quantitatively analyze media with periodic structures. Within the framework of the theory (see, e.g., [9, 14]), the behavior of a heterogeneous media is described with the help of a certain homogenized problem, which is typically a boundary value problem with smooth coefficients, and the solution of a specially constructed problem with periodic boundary conditions. It has been proved that the functions reconstructed by this procedure converge to the exact solution as the cell size ε tends to zero. Moreover, known a priori error estimates qualify the convergence rate in terms of ε. The goal of this paper is to derive two-sided estimates of the modeling error generated by homogenization. In other words, we wish to estimate the difference between the exact solution of the original problem and its approximation obtained by the corresponding homogenized model.

Let Ω⊆ℝd be a bounded domain with Lipschitz boundary ∂Ω, such thatΩ=UiΠiε, where

∏iε=xi+ε∏⌢={x∈ℝd|x−xiε∈Π⌢}

is the basic ‘cell’ (repeating element of the periodic structure, see Fig. 1), which is a simply connected domain with Lipschitz boundary, x_i is the reference point of Πiε, and ∊ is a small parameter (geometrical size of a cell). Here and later on, x denotes the global (Cartesian) coordinate system in ℝd and i= (i₁, i₂. . . i_d) denotes the counting multi-indices for the cells. The notations ∪_i and ∑_i are shorthands for the union and summation over all cells. It is assumed that the overall amount of Πiε in Ω is bounded from above by the quantity

(1.1)c0ε−d, c0=O(1)

and the diameter of Πiε, satisfies the relation

(1.2)diamΠiε =ρε

where ρ is a parameter depending on the geometry of the cell. Usually, ρ is easy to find (e.g., for a cubic cell ρ=d).

Figure 1.

Periodic structure (left) and its basic cell (right).

In the basic cell (see Fig. 1), we use local Cartesian coordinates y ∈ Rd. For any Πiε, local and global coordinates are joined by the relation

y=x−xiε∈Π⌢∀x∈Πiε,∀i.

On Π^, we define a matrix function where ℝsymd×d denotes the set of symmetric d × d matrices. We assume that

(1.3)c1|ξ|2≤A^(y)ξ⋅ ξ≤c2|ξ|2∀ξ∈ℝd,∀y∈Π⌢

where 0 < c₁≤ c₂ < ∞ and introduce the ‘global’ matrix

(1.4)Aε(x):=A^(x−xiε)∀x∈Πiε,∀i

which defines the periodic structure on Ω. In view of (1.3), A_ε (and its inverse counterpart Aε−1) satisfy similar two-sided estimates for any ε.

Consider the second-order elliptic equation

(1.5)− div (Aε∇uε) =fin Ω,f∈L2(Ω)

with homogeneous Dirichlet boundary conditions. The corresponding generalized solution u_ε ∈ H01(Ω) is defined by the relation

(1.6)∫ΩAε∇uε.∇w=∫Ωfw∀w∈H01(Ω).

For any ε > 0, the solution u_ε exists and is unique.

For a function ζ ∈ L¹(ω), where ω is a measurable subset of Ω, we define the mean value by

(1.7)〈ζ〉ω:=1|ω|∫ωζ.

If no confusion may arise, we omit in integrals the symbol of the corresponding Lebesque measure (e.g., dx).However, we write the measure explicitly if it is necessary to distinguish between integration over the global and local coordinates (as in Lemma 2.1).

If we write∫ω〈ζ〉ω, then the average is considered as a constant function on ω (for vector-valued functions, we apply this definition component wise). The error caused by the averaging (1.7) is denoted by

δωζ:=‖ζ−〈ζ〉ω‖ω

where ∥·∥_ω denotes the standard L²-norm on ω.

For a vector (μi)i=1d∈(ℝ>0)d and s ∈ ℝ, μ^s denotes the component wise application of the power s, i.e., μ^s = (μis)i=1d. For vector-valued functions ζ=(ζk)k=1d ∈ L¹ (ω, ℝ^d). and φ:= (φk)k=1dεL¹(Ω,ℝ^d)., we define the local and piecewise constant averages by means of the relations

δωζ:=(‖ζk−〈ζk〉ω‖ω)k=1d, δΩpwφ:=εd/2(∑i‖φk−〈φk〉Πiε‖Πiε)k=1d

and

(δωζ)2:=(‖ζk−〈ζk〉ω‖ω2)k=1d, (δΩpwφ)2:=εd/2((∑i‖φk−〈φk〉Πiε‖Πiε)2)k=1d.

Within the framework of homogenization theory, an approximation of u_ε is constructed by the following procedure (see, e.g., [7, 9, 14]). First, we define (for k = 1, 2 ... d) the solutions N_k of ‘cell problems’

(1.8)div(A^∇Nk)=(divA^)kin Π^ Nk is periodic in Π^∫Π^Nk =0

With the help of them, the homogenized matrix

(1.9)A0=〈Α^(I−∇N)〉Π^

is defined. The function u₀ ∊ H01(Ω) such that

(1.10)∫ΩA0∇u0.∇w=∫Ωfw∀w∈H01(Ω)

provides a “coarse” approximation of u_ε. It is known that (see, e.g., [7]),

uε→u0in L2(Ω),uε→u0in H01(Ω)for ε→0.

However, it is necessary to construct a sequence of more accurate approximations, which converges in a stronger sense. For this purpose, the homogenization theory suggests to use advanced approximations

(1.11)wε1(x):=u0(x)−εψε(x)Nk(x−xiε)∂u0(x)∂χk ∀x∈Πiε,∀i

Where ψ^∊:= min{1, dist(x, ∂Ω)/∊} is a cutoff function.

To prove optimal a priori convergence rates for the modeling error

(1.12)eεmod:=uε−wε1

we need some extra assumptions (see [14], p.28), namely,

(1.13)u0∈W2,∞(Ω¯)

and

(1.14)∂Nk∂yj∈L∞(Π^)

Then, it can be proved (see, e.g., [7], Remark 5.13; [10]; [14], p. 28) that the modeling error satisfies the asymptotic estimates:

(1.15)‖uε−wε1‖H1(Ω)≤c˜ε

and

(1.16)‖Aε∇uε−v0−εv1‖≤c^ε

where

(1.17)v0:=(I−curlyN˜)μμ:=〈A0−1(I−curlyN˜)〉Π˜−1∇u0v1:=curlx(N˜μ)

the d × d matrix N˜with columns N˜kis the solution of the auxiliary problem

(1.18)curlA0−1(curlNk˜(y))=curl(A0−1)k in Π^div N˜k=0N˜k is periodic in Π^∫Π^N˜k=0

and the columns of the matrixcurlyN˜ are given by curlyN˜k, k = 1, 2, …, d.

Numerical methods for homogenized problems are actively studied. Such questions as adaptively and error indication are among the most important questions arising in quantitative analysis of periodical structures. Here, we first of all mention residual type error indicators that develop the ideas suggested in [2, 3] for definite element approximations. Since our approach is based on a different technique, we will sketch here only briefly some relevant literature on residual based estimation and refer for a detailed review, e.g., to [13]. A posteriori error estimates for the heterogeneous multiscale discretization (HMM) of elliptic problems in a periodic setting can be found in [12, 17]. In [1], an a posteriori estimate of residual type for general, possibly non-periodic, diffusion tensors with micro-scales is presented while a residual-type a posteriori error estimate for more general diffusion tensors has been developed in[13]. Also, we mention the papers [4, 5, 29, 30], which are closely related to the topic.

Our goal is to deduce estimates of eεmod of a different type, which provide guaranteed and fully computable bounds of the modeling error. The corresponding error majorant uses the solution of the homogenized problem and, in addition, involves free functions and a function η defined on the cell of periodicity. This freedom can be utilized for improving the efficiency of the corresponding error bounds. Besides, the functions obtained in this way provide efficient reconstructions of the flux. In general, the estimates have the form

(1.19)M⊖(wε1;Θ)≤‖∇(uε−wε1)‖Aε≤M⊕(wε1; η,λ,s)

where

(1.20)‖q‖Aε:=(∫Ω∫Aεq.q)1/2.

The majorant ℳ_⊕ and a minorant ℳ_⊖. are derived in Sections 2 and 3, respectively. Numerical tests are exposed in Section 4. They confirm the efficiency of the estimates.

2 Upper bound of the modeling error

First, we prove a subsidiary result, which states an upper bound of the L²-product of a globally defined function and a periodic function defined on the cell.

Lemma 2.1.

For all g ∈ L²(Ω)^d, η ∈ L²(Π^)^d, and all λ=(λd)k=1d∈(ℝ >0)d it holds

(2.1)∑i∫Πiεg(x)⋅η(x−xiε)dx≤|Ω|〈g〉Ω⋅〈η〉Π^+λ2(δΩpwg)2+λ−12(δΠ^η)2.

Proof. For any g ∈ L²(Ω)^d, we have

J:=∑i∫Πiεg(x).η(x−xiε)dx=∑k=1d∑i∫Πiεgk(x)ηk(x−xiε)dx=∑k=1d∑i∫Πiε(gk(x)−〈gk〉Πiε)ηk(x−xiε)dx+∑k=1d∑i∫Πiε〈gk〉Πiεηk(x−xiε)dx.

Since

(2.2)∑i∫Πiε〈gk〉Πiεηk(x−xiε)dx=εd(∫Π^ηk(y)dy)∑i1|Πiε|∫Πiεgk(x)dx=εd(∫Π^ηk(y)dy)∑i1εd|Π^|∫Πiεgk(x)dx=∫Π^ηk(y)dy1|Π^|∫Ωgk(x)dx

and for any (ck)k=1d∈ℝd

(2.3)∑i∫Πiε(gk(x)−〈gk〉Πiε)ηk(x−xiε)dx=∑i∫Πiε(gk(x)−〈gk〉Πiε)(ηk(x−xiε)−ck)dx≤(∑i‖gk−〈gk〉Πiε‖Πiε)(∫Πiε(ηk(x−xiε)−ck)2dx)1/2=(∑i‖gk−〈gk〉Πiε‖Πiε)εd/2‖ηk−ck‖Π^

we find that

J≤∑k(|Ω|〈gk〉Ω〈ηk〉Π^+(δΩpwg)k‖ηk−ck‖Π^)≤∑k(|Ω|〈gk〉Ω〈ηk〉Π^+λk2(δΩpwg)k2‖ηk−ck‖Π^)=|Ω|〈g〉Ω⋅〈η〉Π^+12λ(δΩpwg)2+12∫Π^∑i1λk(ηk−ck)2dy

for any arbitrary vector λ∈ℝ>0d. In particular, we set ck=〈ηk〉Π^ and obtain (2.1). □

In order to present the main estimate in a transparent form, we introduce the function

(2.4)gτ0(x): =Aε∇wε1−τ0

where

(2.5)τ0∈H(Ω,div):={ϑ∈(L2(Ω))d,divϑ∈L2(Ω)}

and the quantity

(2.6)F(wε1;τ0,η,λ,s):=‖gτ0‖Aε−12+2ε2|Ω|〈gτ0〉Ω.〈η〉Π^+εs(λ−1.(δΠ^η)2+λ⋅(δΩpw(gτ0))2)+c0ε2s‖η‖A^−1,Π^2

where λ∈ℝ>0d, s ∈ ℝ_>0, and

(2.7)η∈H0(Π^, div):{ϑ∈H(Π^,div),〈divϑ〉Π^=0}.

Now, we can deduce the first (general) form of the majorant ℳ_⊕. It is presented in Theorem 2.1 (see also [25]), which proof uses the technique developed in [19–27].

Theorem 2.1.

Let the cell of periodicity Π^ be convex and the conditions (1.1), (1.3), (1.9), (1.11), and(1.13)besatisfied. Then, for any λ ∈ ℝ>0d, s ∈ ℝ_>0τ₀ ∈ H(Ω, div) and η∈ H₀(Π^, div) we have the estimate

(2.8)‖∇(uε−wε1)‖Aε≤M⊕(wε1;τ0,η,λ,s): =F1/2(wε1;τ0,η,λ,s)+C˜FΩ‖divτ0+f‖+εsC˜‖div η‖Π^

where ℱ, C˜FΩandC˜are defined by(2.6)and(2.13), respectively.

Proof. For any v, w∈H01(Ω) and τ ∈H (Ω, div), we have

(2.9)∫ΩAε∇(uε−v)⋅∇w=∫Ω(−Aε∇v⋅∇w+fw)=∫Ω(τ−Aε∇v)⋅∇w+∫Ω(div τ+f)w.

We set w = u_ε − v and estimate the first term in (2.9) as follows:

(2.10)∫Ω(τAε∇v)⋅∇(uε−v)≤‖∇(uε−v)‖Aε‖Aε∇v−τ‖Aε−1.

Henceforth, we select τ in a special form, namely,

(2.11)τ(x)=τ0(x)−εsη(x−xiε) on Πiε

where

η∈H0(Π^, div).

Since

div τ(x)=div τ0(x)−εsdiv η(x−xiε) ∀x∈Πiε,∀i

and

〈div η(⋅−xiε)〉Πiε=εd−1〈div η〉Π^=0

we obtain

∫Ω(div τ+ f)(uε−v)dx=∫Ω(div τ0+ f)(uε−v)dx−∑i∫Πiεεsdiv η(⋅−xiε)(uε−v)dx≤CFΩ‖div τ0+ f‖‖∇(uε−v)‖+εs∑iεd/2−1‖div η‖Π^CΠiε‖∇(uε−v)‖Πiε

where CFΩ is a constant in the Friedrich’s inequality for Ωand CΠiε is a constant in the Poincare’s inequality for Πiε. It is known (cf. [18]) that for convex Πiε

CΠiε≤diam Πiεπ ∀d≥1.

We use (1.1) and (1.2) and arrive at the estimate

∫Ω(div τ + f)(uε−v)dx≤CFΩ‖div τ0 + f‖‖∇(uε−v)‖+εsεd/2−1‖div η‖Π^c0εd/2−1εϱπ‖∇(uε−v)‖=CFΩ‖div τ0 + f‖‖∇(uε−v)‖+εsϱπc0‖div η‖Π^‖∇(uε−v)‖.

In view of (1.3), we obtain

(2.12)∫Ω(div τ + f)(uε−v)≤C˜FΩ‖div τ0 + f‖‖∇(uε−v)‖Aε+εsC˜‖div η‖Π^‖∇(uε−v)‖Aε

where

(2.13)C˜FΩ: =CFΩc1, ϱπc0c1.

Now (2.9), (2.10), and (2.12) imply the estimate

(2.14)‖∇(uε−v)‖Aε≤‖Aε∇v−τ0+εsη‖Aε−1+C˜FΩ‖div τ0 + f‖+εsC˜‖div η‖Π^.

Consider the first term in the right-hand side of the estimate (2.14). We have

‖Aε∇v−τ0+εsη‖Aε−12=∑i∫ΠiεA^−1(x−xiε)(A^(x−xiε)∇v(x)−τ0(x)+εsη(x−xiε))×(A^(x−xiε)∇v(x)−τ0(x)+εsη(x−xiε))dx.

We set v=wε1 and obtain with the help of (2.4)

‖Aε∇wε1−τ0+εsη‖Aε−12=∑i∫Πiε(ε2sA^−1(x−xiε)η(x−xiε)⋅η(x−xiε)+2A^−1(x−xiε)εSgτ0(x)⋅η(x−xiε)+A^−1(x−xiε)εSgτ0(x)⋅gτ0(x))dx.

Now we apply Lemma 2.1 to the second term in the right-hand side of the above relation and arrive at (2.8). □

We note that the estimate (2.14) also holds in a more general setting and can be applied to any reconstruction v (including numerical one) of u_∊ with the requirement that v ∈ H¹(Ω).

Remark 2.1.

It is not difficult to show that the majorant has the same convergence rate as the a priori estimate (cf.(1.16)) provided that the parameters are selected as is recommended by the theory [14].

Indeed, let us choose

(2.15)τ0: =v0−εv1*

where v₀and v₁ are defined by (1.17) and * means periodification of a function,i.e.

w*(x): =w(x,x−xiε)

for anyx∈Πiε and for any i. Then,

div τ0= divv0–ε div v*1= div v0–ε((divx v1)*+ε−1(divy v1)*) .

Sincev1: =−curlx(N˜μ), (cf. (1.17)), the first term in the brackets vanishes and for the second one we use the fact that

(divyv1)*=f+ divxv0

(see, e.g., [7], p. 65). Then, we obtain

(2.16)div τ0= div v0−f− div v0= −f.

Therefore,

(2.17)‖∇(uε−wε1)‖Aε≤M⊕(wε1, τ0,η,λ, s)=F1/2(wε1; τ0,η,λ, s)+εsC˜‖div η‖ ∏^

where ℱ is defined by (2.6) and

gτ0(x)=Aε∇wε1−(v0−εv1).

Then, with the help of (1.15), (1.16), and the triangle inequality, we find that

‖gτ0(x)‖=‖Aε∇(wε1−uε+uε)−(v0−εv1)‖Aε−1 ≤‖Aε∇(wε1−uε)‖Aε−1+‖Aε∇uε−(v0−εv1)‖Aε−1≤cε.

We set η = 0, tend all components of λ to zero and find that

(2.18)ℳ⊕≤cε1/2.

It is worth noting that in some special cases this asymptotic result can be proved in a simpler way. For example, if

A0=〈A^−1〉Π^−1

(which is always the situation in the one-dimensional case or if curl A^−1 = 0), then the simplest choice

τ0= A0∇u0

implies div τ₀ = −f. In this case,

(2.19)∥∇(uε−wε1∥Aε≤ℳ⊕(wε1,τ0,η,λ,s):=ℱ1/2(wε1;τ0,η,λ,s)+εsC˜∥divη∥Π^

where ℱ is defined by (2.6) and

gτ0(x)=Aε∇wε1−A0∇u0

for all y∈Π^, x∈Πiε. Choosing again η = 0 in (2.19), we obtain (2.18).

Remark 2.2.

The right-hand side of the majorant (2.8) is the sum of three non-negative terms, which include a global function τ₀ and a function η defined on the cell of periodicity. This reflects the specifics of the considered class of problems. Hence, the computation of the majorant is based on the flux of the homogenized solution and a proper selection (cf. Section 4) of the function τ defined on the cell of periodicity. The scalar parameters λ_i and the power s can be selected in order to minimize the overall value of the majorant. We emphasize that the computation of the majorant does not require an approximation of the flux associated with the original (global) periodic problem.

The choice

τ0=A0∇u0, η=0

leads to the simplified error estimator

(2.20)‖∇(uε−wε1)‖Aε≤|∑i∫ΠiεA⌢−1(x−xiε)gτ0(x)⋅gτ0(x)dx|1/2=‖Aε∇wε1−A0∇u0‖Aε−1=:M⊕(wε1,u0).

It is easy to show that this simplified majorant is equivalent to the combined primal–dual norm

(2.21)[uε−wε1,Aε∇uε−τ0]: =‖∇(uε−wε1)‖Aε+‖Aε∇uε−τ0‖Aε−1

Indeed, from one hand

(2.22)[uε−wε1,Aε∇uε−τ0]: =‖∇(uε−wε1)‖Aε+‖Aε∇uε−A0∇u0‖Aε−1≤‖∇(uε−wε1)‖Aε+‖Aε∇uε−Aε∇wε1‖Aε−1+‖Aε∇wε1−A0∇u0‖Aε−1≤3‖Aε∇uε−A0∇u0‖Aε−1=3M⊕(wε1,u0).

From the other hand,

(2.23)M⊕(wε1,u0)=‖Aε∇wε1−τ0‖Aε−1≤‖Aε∇wε1−Aε∇uε‖Aε−1+‖Aε∇uε−τ0‖Aε−1=[uε−wε1,Aε∇uε−τ0].

Hence, we obtain

(2.24)M⊕(wε1,u0)≤[uε−wε1,Aε∇uε−τ0]≤3M⊕(wε1,u0).

We note that this result is similar to that has been obtained in [22] for errors of mixed approximations of elliptic partial differential equations.

Remark 2.3.

One can show that in the one-dimensional case, (2.20) holds as equality provided that

(2.25)∫Ω(Aε−1∫0xf)=∫Ω(A0−1∫0xf).

Remark 2.4.

In certain cases, we may know only numerical approximations to the solutions N_k, N˜k and u₀of the auxiliary cell problems (cf.(1.8), (1.18)) and of the homogenized equation (cf. (1.10)). The corresponding approximation errors can be estimated by error majorants of similar types (see [19–24] and references therein). Then, the overall error majorant will include both, approximation and modeling errors. A combined modeling-discretization strategy is suggested in [24] (where the modeling error is generated by defeaturing of a complicated structure) and in [28] (where the modeling error is generated by dimension reduction) and should be used in this case. This topic deserves a separate investigation and lies beyond the framework of this paper which is focused on the principal structure of the guaranteed error bound for homogenized problems.

3 Lower bound of the modeling error

Lower bounds of the modeling error allow us to estimate numerically the sharpness of the error majorant and to evaluate the efficiency of error estimation. A lower bound of the energy error norm can be derived by means of the well known relation (see, e.g.,[22], pp. 85–86):

(3.1)‖∇(uε−v)‖Aε2=supw∈H01(Ω)M⊖2(v;w) : =supw∈H01(Ω)∫Ω(2(f w−Aε∇v⋅∇w)−Aε∇w⋅∇w).

Clearly, for any w∈H01(Ω) it holds ‖∇(uε−v)‖Aε≥ℳ⊖(v;w) Moreover, there exists a function w such that the inequality holds as equality. use (3.1) with v=wε1 (cf. (1.11)) and represent w in the form w = ρ_maxz, where z∈H01(Ω)is a certain specially selected function and the multiplier ρ_max is defined by the relation

ρmax=∫Ω(fz−Aε∇wε1⋅∇z)∫ΩAε∇z⋅∇z

In this case, M⊖2(wε1;ρz) attains its maximum as a quadratic function with respect to ρ. Inserting this value into M⊖2(v;w), we obtain the following lower bound of the modeling error ‖∇(uε−wε1)‖Aε.

(3.2)M⊖2(wε1;z): =|∫Ω(fz−Aε∇wε1⋅∇z)|‖∇z‖Aε=|∫Ω(A0∇u0−Aε∇wε1)⋅∇z|‖∇z‖Aε.

Below we consider two possible choices of the function z. Let

(3.3)z(x): =wε1(x)−u0(x)−ε Θ(x−xiε)

where Θ((x− xi)/∊) is a periodic function defined in Π^, and

(3.4)φε0(x,x−xiε): =−ψε(x)N(x−xiε)⋅∇u0(x).

Then, we rewrite (1.11) in the form

(3.5)wε1=u0+εφε0

and

(3.6)z(x)=ε(φε0(x,x−xiε)−Θ(x−xiε))

we see that in this case the test function z is a periodical function. The minorant is defined by the relation

(3.7)ℳ⊖2(wε1;Θ)=(∫Ω[q⋅∇φε0−q⋅∇Θ])2∫ΩAε∇(φε0−Θ)⋅∇(φε0−Θ)

where

(3.8)q: =(A0−Aε)∇u0−εAε∇φε0.

For this ansatz, the best lower bound will be obtained if (3.7) is maximized with respect to the cell based function Θ and global function ψ_∊. However, in general, finding these (optimal) functions may require essential computational efforts. In the tests below, we used a much simpler choice, namely, Θ = 0,

(3.9)Θ=0, ψε=min{1,1εdist(x,∂Ω)}

and the minorant (3.7) is reduced to

(3.10)ℳ⊖per(Wε1;0)=|∫Ωq⋅∇φε0|‖∇φε0‖Aε.

Also, we may try to find a suitable z represented aperiodically, for example in the form u₀ plus small quasi-periodical disturbances

(3.11)z(x)=ρ(u0(x)+εψεΘ(x−xiε)).

In this case,

(3.12)ℳ⊙2(wε1,Θ)=(∫Ω[q˜⋅∇u0+q˜⋅∇(εψεΘ)])2∫ΩAε∇(u0+εψεΘ)⋅∇(u0+εψεΘ),

where

(3.13)q˜:= (A0 −Aε)∇u0 +ε Aε∇(ψεN∇u0).

In general, the minorant should be maximized with respect to Θ. However, even the simplest choice Θ = 0 yields a lower bound

(3.14)ℳ⊖aper(Wε1;0)=|∫Ωq⋅∇u0|‖∇u0‖Aε.

4 Numerical experiments

A general strategy of computing the majorant consists of minimizing ℳ_⊕ with respect to parameters λ, s, vector function τ ϵ H (Ω, div) and vector function η ϵ H₀(Π^, div) using finite dimensional subspaces S_h (Ω) ⊂ H(Ω, div) (e.g. a finite element space) and S_h(Π^)⊂ H₀(Π^, div), respectively. The process can be started with

(4.1)τ0=A0∇u0,η= 0.

In the numerical experiments discussed below, we set τ and η in accordance with (4.1) and use the simplest error estimator M_⊕(wε1,u₀):

(4.2)‖∇(uε−Wε1)‖Aε≤|∑i∫ΠiεA⌢−1(x−xiε)gτ0(x)⋅gτ0(x)dx|1/2

where g_>τ₀ (x) is defined by (2.4). In most cases, this choice was enough in order to have sufficiently sharp estimates. This is explainable because if the periodic structure is fine and contains many cells, then the correction term is less significant and its influence can be diminished by increasing values of s. However, if a periodic structure is rather coarse (e.g., 25–50 cells) and/or the coefficient of the matrix A^ have jumps, sharp oscillations, etc. then the term 𝜀^s𝛈 may augment the homogenized flux substantially and it may be required to use the most general form of the majorant.

Below, we apply the estimates derived in Sections 2 and 3 to several one- and two-dimensional test problems. For this purpose, we select problems used in publications related to analysis of homogenized and interface problems, e.g., see [8, 13, 15, 16, 30]. Our goal is to validate the sharpness of the two-sided error bounds presented by M_⊕. and two lower bounds introduced in Section 3 (i.e.,M_⊕is computed by ℳ⊖per(Wε1;0) or ℳ⊖per(Wε1;0); (3.10) and (3.14)).

For the quantitative characterization of two-sided bounds, we use the number

(4.3)x:=M⊕M⊖

which can be also viewed as a computable upper bound of the efficiency index

i⊕eff:=M⊕‖∇(uε−wε1)‖Aε

and gives insights of the quality of the error majorant. Similarly, we define the efficiency index of the lower bound

iΘeff:=MΘ‖∇(uε−wε1)‖Aε.

In the first series of tests, we set d=1 and Ω=(0, 1). Then, uε∈H01(Ω) is defined by the relation

(4.4)∫01Aεu′εv′=∫01fv ∀v∈H01(Ω)

Example 4.1.

Let

A^(y):={1, 0<y≤1/22, 1/2<y<1

and A_ε is defined as in (1.4). The right-hand side is given by f:= sin (2πx/ε). Here, the explicit forms of A₀, u′0, dN/dy and N are known (they can be found from (1.9) and (1.8)):

A0(x)=43u′0=3ε8πcos(2πxε−1)dNdy(y)={−13, 0<y≤1/213, 1/2<y<1N(y)={−y3+112, 0<y≤1/2y3−14, 1/2<y<1

Example 4.2.

Let A_ε(x) = 2 + cos (2πx/ε), f := e^{10 x}. Then, see (1.9) and (1.8):

A0(x)=3u′0=−3−0.510e10x+3−0.5100e10dNdy(y)=1−3(2+cos(2πy))−1N(y)=∫(1−3(2+cos(2πy))−1)dy.

In Example 4.1,f is a periodic function. Therefore, it is natural to expect that the minorant M⊖per (in which the periodicity is taken into account) will provide better results. In Example 4.2, the right-hand side is represented by a non-periodical function, and, therefore, we expect that M⊖aper will be better (at least for problems with relatively small amount of cells). The corresponding numerical results are depicted in Fig. 2 and confirm the proposed choice of the lower error bound. We note that in Example 4.1 the equality (2.25) holds and (cf. Remark 2.3) the majorant (4.2) coincides with the error. This fact is con_rmed numerically (see Fig. 2a, 2b). Example 4.2 shows that the majorant and minorants are quite sharp if the number of cells is sufficiently large (regardless of the condition (2.25)).

Figure 2.

Error bounds (left) and efficiency indices (right) for Example 4.1 and Example 4.2.

Example 4.3.

Let d = 2, Ω = (0, 1)², and uε ∈ H01(Ω) be defined by the relation

∫ΩAε∇uε.∇v=∫Ωfv ∀v∈H01(Ω)

Here A_𝜀 is generated by the matrix A^:=aI (cf. (1.4)), where

(4.5)a:={a1>0 in(0,12)2∪(12,1)2a2>0 in (0,1)2\ (0,12)2∪(12,1)2. Π^:a2a1a1a2

Then (see, e.g.,[14], pp. 35–39), A0=a1,a2 We choose

f= 2a1a2 (x1(1 −x1) +x2(1 −x2))

such that

(4.6)u0(x) =x1x2(1 −x1)(1 −x2).

Exact solutions of the cell problems

(4.7)∂∂yi(A^ij(y)∂Nk(y)∂yj)=∂∂yiA^ik(y) inΠ^=(0,1)2Nk is periodic in Π^〈Nk〉Π^=0

(4.8)Nk(y)=v(y+12)+yk,k=1,2.

Here v(y) is the unique solution of the problem

(4.9)− div (a.∇v) = 0 in (−1, 1)2

with homogenous Dirichlet boundary conditions and a is defined by (4.5). This solution is given in polar coordinates (r, ℐ) centered at the origin by the relation

(4.10)v=rγμ(ϑ)

where

(4.11)μ(ϑ): ={cos(αβ)cos((ϑ−π2+α)γ),0≤ϑ≤π2cos(αγ)cos((ϑ−π+β)γ),π2≤ϑ≤πcos(αβ)cos((ϑ−π−α)γ),π≤ϑ≤3π2cos((π2−α)γ)cos((ϑ−3π2−β)γ),3π2≤ϑ≤2π

is a continuous and piecewise smooth function and the numbers α, β, and γ depend on a₁/a₂ and satisfy the relations

(4.12){a1a2=tan(βγ)cot(αγ)a2a1=tan(αγ)cot(βγ)a1a2=−tan(βγ)cot((π2−α)γ)a2a1=tan((π2−α)γ)cot(βγ)γ>0max{0,πγ−π}<2αγ<min{πγ,π}max{0,π}<−2βγ<min{π,2π}

It is known that v has a restricted regularity (namely,v∈H1+γ−ε(Π^) for any ∊ > 0).

We use this fact in order to verify the efficiency of the error majorant in different situations, we consider two cases, in which the ratio between a₁ and a₂ (and the regularity of N_k) are quite different.

Case 1: let a₁ = 5.0, a₂ = 1.0. In this case, the solution (4.9) has γ = 0.53544094560 and ℐ = π/2 (cf. system (3.2) in [15]) so that v ∈ H^3/2(Ω).
Case 2: now, we set γ = 0.1 and ℐ = π/2. By solving (4.11) and (4.12), we find that in this case a₁=161.4476387975881 and a2 = 1.0. Here, v∈H1+α(Π^).) with 0 < α < 0.1, i.e., it is almost an H¹ function.

To quantify the efficiency of the estimates (4.2) and (3.10), we compare them with the exact error

(4.13)e: =‖∇(uε−wε1)‖Aε

Since u_∊ is unknown, we replace it by the ‘reference’ solution u_ref computed on a very fine mesh (h ≪ ∊). The corresponding efficiency indices are defined by the relations

(4.14)i⊕eff=M⊕(wε1;0,1,1)‖∇(uref−wε1)‖Aε,i⊕eff,per=M⊖per(wε1;0)‖∇(uref−wε1)‖Aε

In Table 1 (Case 1) and 2 (Case 2), we present these quantities together with the quantity x as in (4.3). We see that the estimates adequately reproduce the modeling error.

Table 1

Efficiency of error majorant and minorant for Example 4.3, Case 1.

ε⁻¹	i⊕eff	i⊖eff	x
8	1.0714	0.8824	1.2141
16	1.0874	0.8781	1.2384
32	1.0988	0.8591	1.2790
64	1.1633	0.8461	1.3749

Table 2

Efficiency of error majorant and minorant for Example 4.3, Case 2.

ε⁻¹	i⊕eff	i⊖eff	x
8	1.0724	0.8291	2.0533
16	1.9701	0.7961	2.4750
32	2.1848	0.7370	2.9644
64	2.2771	0.7124	3.1964

It is quite predictable that the estimates are better in the first case (related to a more regular v ). For the first problem, efficiency indices of the majorant and minorant are quite close to 1. However, the estimates are also valid for the second case (minimal regularity). Indeed, the efficiency index of the majorant does not exceed 2.3 and the one of the minorant does not go below 0.7.

Acknowledgment

The authors are grateful to Swiss National Science Foundation for supporting this research under the grants 200021_119809 and 200020_134621. The first co-author also thanks the Institute for Mathematical Research (FIM, ETH, Zurich) for support. We are grateful to Dr. Christian Wüst — the finite element computations have been performed on the basis of his program JCFD.

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Received: 2014-1-26

Revised: 2014-11-21

Accepted: 2015-1-28

Published Online: 2016-3-14

Published in Print: 2016-3-1

Estimates of the modeling error generated by homogenization of an elliptic boundary value problem

Abstract

1 Introduction

2 Upper bound of the modeling error

3 Lower bound of the modeling error

4 Numerical experiments

Acknowledgment

References

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