## 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.

## 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

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

**x**denotes the global (Cartesian) coordinate system in

**i**= (

*i*

_{1},

*i*

_{2}. . .

*i*) denotes the counting multi-indices for the cells. The notations ∪

_{d}_{i}and ∑

_{i}are shorthands for the union and summation over all cells. It is assumed that the overall amount of

*Ω*is bounded from above by the quantity

and the diameter of

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

In the basic cell (see Fig. 1), we use local Cartesian coordinates

On *d* × *d* matrices. We assume that

where 0 < *c*_{1}≤ *c*_{2} < ∞ and introduce the ‘global’ matrix

which defines the periodic structure on *Ω*. In view of (1.3), *A _{ε}* (and its inverse counterpart

*) satisfy similar two-sided estimates for any*A ε − 1

*ε*.

Consider the second-order elliptic equation

with homogeneous Dirichlet boundary conditions. The corresponding generalized solution *u _{ε}* ∈

For any *ε* > 0, the solution *u*_{ε} exists and is unique.

For a function *ζ* ∈ *L*^{1}(*ω*), where *ω* is a measurable subset of *Ω*, we define the mean value by

If no confusion may arise, we omit in integrals the symbol of the corresponding Lebesque measure (e.g., d**x**).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* (for vector-valued functions, we apply this definition component wise). The error caused by the averaging (1.7) is denoted by

where ∥·∥_{ω} denotes the standard *L*^{2}-norm on *ω*.

For a vector *s* ∈ ℝ, *μ*^{s} denotes the component wise application of the power *s*, i.e., *μ*^{s} = *L*^{1} (*ω*, ℝ^{d}). and **φ**:= *ε**L*^{1}(Ω,ℝ^{d})., we define the local and piecewise constant averages by means of the relations

and

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*of ‘cell problems’

_{k}With the help of them, the homogenized matrix

is defined. The function *u*_{0} ∊

provides a “coarse” approximation of *u _{ε}*. It is known that (see, e.g., [7]),

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

Where *ψ ^{∊}*:= min{1, dist(

**x**,

*∂Ω*)/

*∊*} is a cutoff function.

To prove optimal *a priori* convergence rates for the modeling error

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

and

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

and

where

the *d* × *d* matrix

and the columns of the matrix*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 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

where

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 *L*^{2}-product of a globally defined function and a periodic function defined on the cell.

For all **g** ∈ *L*^{2}(*Ω*)^{d}, **η** ∈ *L*^{2}(^{d}, and all

*Proof*. For any **g** ∈ *L*^{2}(Ω)^{d}, we have

Since

and for any

we find that

for any arbitrary vector and obtain (2.1). □

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

where

and the quantity

where ,

*s*∈ ℝ

_{>0}, and

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].

Let the cell of periodicity *, and*(1.13)*be**satisfied. Then, for any *,

*s*∈ ℝ

_{>0}

**τ**∈

_{0}*H*(Ω, div) and

**η**∈

*H*

_{0}(

*, div) we have the estimate*Π ^

*where* ℱ, *and**are defined by*(2.6)*and*(2.13), respectively.

*Proof.* For any *v*, *τ* ∈*H* (*Ω*, div), we have

We set *w* = *u _{ε}* −

*v*and estimate the first term in (2.9) as follows:

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

where

Since

and

we obtain

where *Ω*and

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

In view of (1.3), we obtain

where

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

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

We set

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* ∈ *H*^{1}(*Ω*).

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

where **v _{0}**and

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

_{1}for any**i**. Then,

Since

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

Therefore,

where ℱ is defined by (2.6) and

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

We set ** η** =

**0**, tend all components of

*λ*to zero and find that

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

(which is always the situation in the one-dimensional case or if curl **0**), then the simplest choice

implies div **τ**_{0} = −*f*. In this case,

where ℱ is defined by (2.6) and

for all **η** = **0** in (2.19), we obtain (2.18).

The right-hand side of the majorant (2.8) is the sum of three non-negative terms, which include a global function **τ**_{0} 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

leads to the simplified error estimator

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

Indeed, from one hand

From the other hand,

Hence, we obtain

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

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

In certain cases, we may know only numerical approximations to the solutions *N _{k}*,

*u*

_{0}of 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):

Clearly, for any it holds

*w*such that the inequality holds as equality. use (3.1) with

*(cf. (1.11)) and represent*v = w ε 1

*w*in the form

*w*= ρ

_{max}z, where

_{max}is defined by the relation

In this case,

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

where Θ((**x**− **xi**)/∊) is a periodic function defined in

Then, we rewrite (1.11) in the form

and

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

where

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,

and the minorant (3.7) is reduced to

Also, we may try to find a suitable *z* represented aperiodically, for example in the form *u*_{0} plus small quasi-periodical disturbances

In this case,

where

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

## 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 *H*_{0}(*S _{h}* (Ω) ⊂

*H*(Ω, div) (e.g. a finite element space) and

*S*(

_{h}*H*

_{0}(

In the numerical experiments discussed below, we set **τ** and **η** in accordance with (4.1) and use the simplest error estimator *M*_{⊕}(*w*,

*u*

_{0}):

where **g**_{>τ}_{0} (**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 ^{s}*𝛈* 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

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

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

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

In the first series of tests, we set *d*=1 and Ω=(0, 1). Then,

Let

and *A _{ε}* is defined as in (1.4). The right-hand side is given by

*f*:= sin (2

*πx*/

*ε*). Here, the explicit forms of

*A*

_{0},

*, d*u ′ 0

*N*/d

*y*and

*N*are known (they can be found from (1.9) and (1.8)):

Let *A _{ε}*(

*x*) = 2 + cos (2π

*x*/

*ε*),

*f*:= e

^{10 x}. Then, see (1.9) and (1.8):

In Example 4.1,*f* is a periodic function. Therefore, it is natural to expect that the minorant (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

*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)).*M ⊖ aper

Let *d* = 2, Ω = (0, 1)^{2}, and

Here *A _{𝜀}* is generated by the matrix

*(cf. (1.4)), where*A ^ : = a I

Then (see, e.g.,[14], pp. 35–39),

such that

Exact solutions of the cell problems

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

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

where

is a continuous and piecewise smooth function and the numbers α, β, and γ depend on *a*_{1}/*a*_{2} and satisfy the relations

It is known that *v* has a restricted regularity (namely,

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 *a*_{1} and *a*_{2} (and the regularity of *N _{k}*) are quite different.

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

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

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.

ε^{−1} | 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 |

ε^{−1} | 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

© 2016 by Walter de Gruyter Berlin/Boston