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 twosided 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.
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 twosided 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
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
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
Consider the secondorder 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., 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
where ∥·∥_{ω} denotes the standard L^{2}norm on ω.
For a vector
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_{k} of ‘cell problems’
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
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 nonperiodic, diffusion tensors with microscales is presented while a residualtype 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
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.
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}(
Proof. For any g ∈ L^{2}(Ω)^{d}, we have
Since
and for any
we find that
for any arbitrary vector
In order to present the main estimate in a transparent form, we introduce the function
where
and the quantity
where
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
where ℱ,
Proof. For any v,
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
we obtain
where
We use (1.1) and (1.2) and arrive at the estimate
In view of (1.3), we obtain
Now (2.9), (2.10), and (2.12) imply the estimate
Consider the first term in the righthand side of the estimate (2.14). We have
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
for any
Since
implies div τ_{0} = −f. In this case,
where ℱ is defined by (2.6) and
The righthand side of the majorant (2.8) is the sum of three nonnegative 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
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 onedimensional case, (2.20) holds as equality provided that
In certain cases, we may know only numerical approximations to the solutions N_{k},
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
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 quasiperiodical 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
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}(
In the numerical experiments discussed below, we set τ and η in accordance with (4.1) and use the simplest error estimator M_{⊕}(w
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
Below, we apply the estimates derived in Sections 2 and 3 to several one and twodimensional 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 twosided error bounds presented by M_{⊕}. and two lower bounds introduced in Section 3 (i.e.,M_{⊕}is computed by
For the quantitative characterization of twosided 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 righthand side is given by f:= sin (2πx/ε). Here, the explicit forms of A_{0},
Let A_{ε}(x) = 2 + cos (2πx/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
Let d = 2, Ω = (0, 1)^{2}, and
Here A_{𝜀} is generated by the matrix
Then (see, e.g.,[14], pp. 35–39),
Exact solutions of the cell problems
Here v(y) is the unique solution of the problem
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.
The authors are grateful to Swiss National Science Foundation for supporting this research under the grants 200021_119809 and 200020_134621. The first coauthor 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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