Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.


Introduction
In this work we consider second-order elliptic equations of nondivergence structure, involving rapidly oscillating coefficients, of the form Here we assume that Ω ⊂ R is a sufficiently regular bounded domain, > 0 is small, and that = ( ) : R → R × is a symmetric, uniformly elliptic and (0, 1) -periodic matrix-valued function such that ∈ 1, ( ) for some > , where := (0, 1) denotes the unit cell; see Section 2.1. The main goal of this paper is to propose and analyze a numerical homogenization scheme for (1.1), (1.2) that is based on finite element approximations. The theory of periodic homogenization is concerned with the limiting behavior of the solutions as the oscillation parameter tends to zero. For the problem (1.1), (1.2) under consideration a classical homogenization theorem (see [14], Sect. 3, Thm. 5.2) states that the solution sequence ( ) >0 converges in an appropriate Sobolev space to the solution 0 to the problem Here 0 ∈ R × is the constant matrix given by and : R → R is the invariant measure, i.e., the solution to the problem see Section 2 for further details. The task of numerical homogenization is the numerical approximation of the matrix 0 and the solution 0 to the homogenized problem (1.3). As it turns out, 0 provides a good approximation to in 1 (Ω), and by adding corrector terms it is possible to obtain an 2 (Ω)-norm approximation. Note that the approximation of (1.1), (1.2) by a standard 2 (Ω)-conforming finite element method does not yield error bounds independent of , since for > 0 one has that The motivation for investigating second-order elliptic problems in nondivergence-form comes from physics, engineering, as well as mathematical areas such as stochastic analysis. A notable example of a nonlinear PDE of nondivergence structure is the Hamilton-Jacobi-Bellman equation, which arises in stochastic control theory. The asymptotic behavior of PDEs with rapidly oscillating coefficients is also of importance when microinhomogeneous media are investigated.
Over the past decades significant work has been done on periodic homogenization of elliptic problems in divergence-form; numerical homogenization for nondivergence-form problems is however less developed.
The theory of homogenization of divergence-form problems such as in Ω (1.5) with periodic and sufficiently regular : R → R × and : R → R is extensively covered in the books [6,14,18,37]. For divergence-form problems, various multiscale finite element methods (MsFEM) have been developed, which have the advantage over classical finite element methods of providing accurate approximations for very small values of even for moderate values of the grid size. The book [19] by Efendiev and Hou contains a detailed overview of these methods.
It is important to note that although, if is sufficiently smooth, equation (1.1) can be rewritten in divergenceform, in Ω, (1.6) this equation does not fit into the framework of divergence-form homogenization problems such as (1.5), because of the −1 term in front of the first-order term in (1.6). Such diffusion models with large drift have been considered by various authors [7,8,17,20,30,31]; they require either specific assumptions or the resolution of additional computationally onerous spectral problems. For the theory of homogenization of nondivergence-form problems such as (1.1) we refer to the monograph [14] by Bensoussan et al., to the paper [10] by Avellaneda and Lin, and the references therein. In [13], Bensoussan et al. study the more general problem involving a Hamiltonian with quadratic growth. Numerical homogenization for nondivergence-form problems using finite difference schemes has been considered in [22] by Froese and Oberman.
The numerical method presented in this paper has resemblances with the finite element heterogeneous multiscale method (HMM). The HMM has been introduced in [38] by E and Engquist and has been successfully applied to many multiscale problems. For an overview of the field of finite element HMM, we refer to the articles [2][3][4][5] by Abdulle and co-authors, and the references therein. An a priori error analysis for the fully discrete finite element HMM for elliptic homogenization problems in divergence-form can be found in the work [1] by Abdulle. Concerning nondivergence-form problems, a finite difference HMM has recently been used for the numerical homogenization of second-order hyperbolic nondivergence-form problems by Arjmand and Kreiss [9].
The first step in the development of the proposed numerical homogenization scheme is the construction of a finite element method to obtain approximations ( ℎ ) ℎ>0 ⊂ 1 per ( ) to the invariant measure with optimal order convergence rate where˜ℎ denotes the finite-dimensional subspace of 1 per ( ) consisting of continuous -periodic piecewise linear functions on the triangulation with zero mean over ; see Theorem 3.1.
Throughout this work, we use the notation for , ∈ R to denote that ≤ for some constant > 0 that does not depend on and the discretization parameters.
The second step is to obtain approximations ( 0 ℎ ) ℎ>0 ⊂ R × to the constant matrix 0 ; see Lemma 3.3. To this end, the integrand in (1.4) is replaced by its continuous piecewise linear interpolant and the invariant measure is replaced by the approximation ℎ , i.e., which can be computed exactly using an appropriate quadrature rule. The third step is to perform an (Ω)-conforming ( ∈ {1, 2}) finite element approximation for the problem {︃ 0 ℎ : 2 ℎ 0 = in Ω, ℎ 0 = 0 on Ω, on a family of triangulations of the computational domainΩ, parametrized by a discretization parameter > 0, measuring the granularity of the triangulation, to obtain ( ℎ, where the constant is independent of ℎ; see Lemma 3.6. Note that for the sake of approximating 0 , an 1 (Ω)conforming finite element method is sufficient. The approximation ( ℎ, 0 ) ℎ, >0 ⊂ (Ω) ∩ 1 0 (Ω) obtained by this procedure approximates 0 , i.e., the solution to (1.3), with convergence rate which can be improved to (ℎ 2 + ) for more regular ; see Theorems 3.5, 3.9 and Remark 3.10.
Concerning the approximation of , i.e., the solution to (1.1), (1.2), we show in Section 2 that under certain assumptions on the domain and the right-hand side, one has that where the corrector functions : R → R, , = 1, . . . , , are defined as the solutions to This provides us with the estimate which shows that 0 is a good 1 (Ω)-norm approximation to for small , and we show in Sections 3.2 and 3.3 how the above estimate can be used to obtain approximations to 2 . Note that in order to approximate in the 1 (Ω)-norm, it is sufficient to approximate 0 in the 1 (Ω)-norm. However, for an approximation of 2 based on the above corrector estimate, we need to approximate 0 in the 2 (Ω)-norm.
In Section 3.4, we extend our results to the case of nonuniformly oscillating coefficients, i.e., to problems of the form where = ( , ) : Ω×R → R × is a symmetric, uniformly elliptic matrix-valued function that is -periodic in for fixed ∈ Ω, and such that ∈ 2,∞ (Ω; 1, ( )) for some > .
We prove the corrector estimate where 0 is the solution to the homogenized problem corresponding to (1.7) and are certain corrector functions. We then discuss the numerical approximation of based on this corrector estimate; see Section 3.4. In Section 4, we present numerical experiments for problems with periodic and nonuniformly oscillating coefficients, demonstrating the theoretical results.
Finally, in Section 5, we collect the proofs of the results contained in this work.

Homogenization of elliptic problems in nondivergence-form
In this section, we study the homogenization of elliptic problems in nondivergence-form with periodic coefficients. The outline of this section is as follows.
We provide the statement of the problem in Section 2.1, i.e., we define sets of assumptions for the domain, the coefficients and the right-hand side, ensuring well-posedness of the problem. In Section 2.2, we introduce the invariant measure and describe a well-known procedure for transforming the original nondivergence-form problem into a divergence-form problem. This is used in Section 2.3 in combination with uniform 2, estimates to carry out the homogenization for the problem under consideration. Finally, we introduce correctors in Section 2.4 and derive 2, homogenization results.

Framework
We denote the unit cell in R by and consider a symmetric matrix-valued function (2.1) By Sobolev embedding, we then have that ∈ 0, (R ) for some 0 < ≤ 1.
For > 0, we are concerned with the problem where the triple (Ω, , ) satisfies one of the following sets of assumptions.

Transformation into divergence-form
We recall a well-known procedure to transform the problem (2.2) into divergence-form; see [10,14]. We use the notation Let us start by introducing the notion of invariant measure; see [14].
The function is called the invariant measure. We have that ∈ 1, ( ), see [15,16], and there exist constants¯, > 0 such that Moreover, for a -periodic function ∈ 2 (R ), the adjoint problem We note that the function is only in 1, ( ) in general, and in particular it does not belong to 2 ( ), as can be seen from the example chosen in Section 4.1. With the invariant measure at hand, we can easily convert the problem into divergence-form as follows. We define a matrix-valued function = ( ) 1≤ , ≤ : R → R × by with ∈ per ( ) denoting the solution to for 1 ≤ ≤ . Since ∈ 1, ( ) and ∈ 1, ( ), by elliptic regularity one has that ∈ 2, ( ) for any 1 ≤ ≤ . Hence, we have that Further, we observe that is skew-symmetric, -periodic with zero mean over , and that div( ) = −div( ) a.e. on R .
Then, since div( div ) = 0, and using the fact that is skew-symmetric, we obtain that is, we have converted (2.2) into divergence-form: and it is straightforward to check that div is -periodic, Hölder continuous on R and uniformly elliptic.

Uniform 2, estimates and homogenization theorem
The transformation described in the previous section can be used to obtain uniform 2, (Ω) a priori estimates for the solution of (2.2), which are crucial in deriving homogenization results.

Correctors
We show next that by adding corrector terms to the solution 0 of the homogenized problem we obtain a 2, convergence result.

The numerical scheme
In this section, we present and rigorously analyze the proposed numerical scheme. The outline of this section is as follows.
Section 3.1 is divided into three parts and discusses the numerical homogenization. In the first part, we approximate the invariant measure by a finite element method and provide a convergence result for the approximation. This is then used in the second part to obtain an approximation to the effective coefficients, i.e., to the constant matrix 0 . In the third part, we use a finite element method to discretize the homogenized problem and show convergence results for the approximation of the homogenized solution in 1 (Ω) and 2 (Ω), using the approximated effective coefficients, a comparison result, and two technical lemmata. Improvements to the convergence rates are given, provided more regularity on the coefficients is assumed.
In Section 3.2, we address the approximation of the corrector functions, presenting a method of successively approximating higher derivatives. We then use the homogenization results obtained in Section 2 and the approximations of the homogenized solution and the corrector functions from the previous subsections to approximate the original solution in Section 3.3. Finally, we study the case of nonuniformly oscillating coefficients in Section 3.4, derive homogenization results similar to the case of periodic coefficients and discuss the numerical homogenization for this case.

Numerical homogenization scheme
The first step is to approximate the invariant measure.

Approximation of
For the approximation of the invariant measure , we consider a shape-regular triangulation of into triangles with longest edge ℎ > 0 and let }︂ be the finite-dimensional subspace of per ( ) consisting of continuous -periodic piecewise linear functions on the triangulation with zero mean over . We assume that Then we have the following approximation result for .
Then, for ℎ > 0 sufficiently small, there exists a unique˜ℎ ∈˜ℎ such that

1)
and writing ℎ :=˜ℎ + 1, we have that where is the invariant measure, as defined in Lemma 2.4.
as ℎ tends to zero.

Approximation of 0
We use this finite element approximation of the invariant measure to obtain an approximation to the constant matrix To this end, we first replace the invariant measure by the approximation ℎ from Theorem 3.1, and then replace the integrand by its piecewise linear interpolant, This integral can be computed exactly using an appropriate quadrature rule. The following lemma gives an error estimate for this approximation.
be the constant matrix given by Theorem 2.6, let ℎ be the approximation to the invariant measure given by Theorem 3.1, and let 0 ℎ = ( 0 ,ℎ ) ∈ R × be the matrix given by Then, for ℎ > 0 sufficiently small, 0 ℎ is elliptic and

Approximation of 0
For the approximation of the solution 0 to the homogenized problem, we use the following comparison result for the error committed when replacing 0 by 0 ℎ .
Finally, we can use an 1 0 (Ω)-conforming finite element approximation ℎ, 0 to the solution ℎ 0 of (3.2), satisfying the error bound with constants independent of ℎ. By the triangle inequality and the results obtained in this section, we have the following approximation result for 0 .
In the proof of Lemma 3.6, we use the following result on the regularity of solutions to Poisson's problem on convex polygons; see also [26,28,29,32].
(i) Approximation of : In this case, ∈ 2 ( ) and we have that by choosing˜ℎ = ℐ ℎ − ∫︀ ℐ ℎ , and using an interpolation error bound. Therefore, Theorem 3.1 yields (ii) Approximation of 0 : By an interpolation error bound and the fact that ℎ is piecewise linear, one has Therefore, the proof of Lemma 3.3 yields (iii) Approximation of 0 : It follows that the results of Lemma 3.4, Theorems 3.5 and 3.9 can be improved to second-order convergence in ℎ, i.e., We note that second-order convergence with respect to ℎ could not have been obtained by using a piecewise constant approximation of ℎ instead of the piecewise linear approximation considered here. For the approximation of derivatives of 0 of higher than second order, the post-processing method of Babuška [12] can be used to obtain error bounds in norms involving derivatives of higher order than the energy norm (the norm natural to the problem).
For bounded convex polygonal domains Ω ⊂ R 2 , an 2 -conforming approximation to the solution of (3.2) can be obtained as follows. Assume that ∈ 1 0 (Ω) so that (3.3) holds. Consider a shape-regular triangulation of Ω into triangles with longest edge > 0, and let be an appropriate finite element space. In practice, the Hsieh-Clough-Tocher element and the Argyris element can be used as 2 -conforming elements. Then, for ℎ > 0 sufficiently small, standard finite element analysis can be used to show that there is a unique function ℎ, and that the error bound (3.5) holds. Further finite element approaches for approximating the solution of nondivergence-form problems include the conforming method [33] that makes use of a finite element Hessian, the discontinuous ℎ -Galerkin method [35,36], the primal method [21] similar to an interior penalty discontinuous Galerkin method, the mixed finite element method [24], and the variational formulations presented in [23].

Approximation of the corrector
We now address problem (2.8) and present a method for ∈ 2,∞ ( ). To simplify the notation and the arguments, we assume that we know the invariant measure and the matrix 0 = ( 0 ) 1≤ , ≤ exactly instead of working with our approximation 0 ℎ . For a given -periodic right-hand side ∈ 2,∞ ( ), we consider the problem Obtaining an approximation for second-order derivatives via finite elements is not straightforward since the natural solution space is per ( ). We present a method of successively approximating higher derivatives. Let ℎ be a per ( )-conforming finite element approximation to , i.e., with ℎ ⊂ per ( ) finite-dimensional, and satisfying the error estimate Let ∈ {1, . . . , } and write := . Then, using the equation we find that in a weak sense, one has Further, we claim that ∈ per ( ). Indeed, this follows from the regularity and periodicity of and Therefore, ∈ per ( ) satisfies Now we use our 1 -conforming approximation for for the right-hand side and use a per ( )-conforming finite element method for approximating the solution ∈ per ( ) to the following problem: where is such that this problem admits a unique solution (such that the solvability condition (2.5) is satisfied). By looking at the problem for − , one obtains the comparison result Let ℎ be a per ( )-conforming finite element approximation to the solution of (3.7) satisfying for some constant = (‖ ‖ 2,∞ ( ) ) > 0. Then, using the triangle inequality, we obtain for some constant = (‖ ‖ 2,∞ ( ) ) > 0. Using this procedure for = 1, . . . , , we eventually obtain approximations to derivatives of order up to two of .
Then we know that (2.10) is satisfied, and by Theorem 2.8 we have that where 0 is the solution to the homogenized problem, and are the corrector functions given as the solutions to (2.8). This result can be used to construct an approximation of , i.e., to the solution of problem (2.2) for small . We note that (3.8) implies that

(3.9)
This leads to the following approximation result for .
Theorem 3.11 (Approximation of ). In the situation described above, let ( 0,ℎ ) ℎ>0 ⊂ 2 (Ω) be a family of 2 -conforming approximations for 0 satisfying the error bound and for 1 ≤ , , , ≤ , let ( ,ℎ ) ℎ>0 ⊂ 2 per ( ) be a family of 2 approximations for 2 satisfying the error bound Then, by writing we have that In connection with the previously described approximation of the homogenized solution 0 and the corrector functions , note that Theorem 3.9 provides an 2 (Ω)-conforming approximation to 0 and the method presented in Section 3.2 provides 2 per ( ) approximations for the second-order partial derivatives of , as required for the setting of Theorem 3.11.
Let us conclude this section by remarking that if the second derivatives of the corrector functions are approximated in the space ∞ ( ) or if the solution to the homogenized problem is approximated in the space 2,∞ (Ω), then one obtains by a similar proof an approximation result for the second derivatives of in 2 (Ω).
As in Section 2, uniform a priori estimates for the solution to (3.10) allow passage to the limit in equation (3.10); see [13,14]. The coefficient matrix of the homogenized problem now depends on the slow variable , and is obtained by integrating against an invariant measure. Corrector results can then be shown as before.
First, we consider a shape-regular triangulation onΩ consisting of nodes { } ∈ with grid size > 0, and a shape-regular triangulation ℎ on with grid size ℎ > 0. Then, for any ∈ , we can use the scheme from Section 3.1 (see Thm. 3.1) to obtain an approximation ℎ ∈ 1 ( ) to = ( , ·) such that Further, we obtain that 0, is an approximation to 0 ( ) (see Lem. 3.3), (3.14) Now we define 0 ℎ, to be a continuous piecewise linear function on the triangulation such that 0 ℎ, ( ) = 0, ℎ for all ∈ . Then, using (3.14) and denoting the continuous piecewise linear interpolant of a function on the triangulation by ℐ , we have We observe that, similarly to the proof of Lemma 3.4, we obtain that the solution ℎ, satisfies, for ℎ, > 0 sufficiently small, and in view of (3.15), where 0 is the solution to the homogenized problem (3.13). Here we have used the interpolation estimate which follows from 0 ∈ 2,∞ (Ω) and standard interpolation theory.
Remark 3.16. For problems in divergence-form, similar results have been derived previously using heterogeneous multiscale methods; see e.g., [1].
At this point, let us note that in contrast with our procedure of approximating the effective coefficient 0 at the nodes of the coarse triangulation and interpolating linearly, in the framework of the finite element heterogeneous multiscale method 0 is typically approximated at the coarse integration nodes; see e.g., [1,2]. The use of piecewise linear interpolation allows us to obtain second-order convergence. Assuming more regularity on the coefficient ( , ) in , as in Remark 3.10, the error in the approximation of the homogenized solution 0 can be improved to order ( 2 + ℎ 2 ). Finally, the solution to (3.16) can be approximated by a standard finite element method on the triangulation , which yields an approximation 0,ℎ, ∈ 2 (Ω) ∩ 1 0 (Ω) to 0 in the 2 (Ω)-norm.
The approximation of can be obtained based on the corrector estimate from Theorem 3.15 analogously as in Section 3.3.

Numerical experiments
In this section, we illustrate the theoretical results through numerical experiments. We provide an example for the case of periodic coefficients in Section 4.1, and one for the case of nonuniformly oscillating coefficients in Section 4.2. In both cases, we provide not only an example with an unknown 0 , but also a set-up with a known 0 in order to test the approximation scheme for the homogenized solution.
The experiments demonstrate the performance of the scheme for the approximation of the invariant measure , the effective coefficients 0 , the homogenized solution 0 , as well as the approximation of the solution to the original problem for a fixed value of .

Periodic coefficients
We consider the homogenization problem with the matrix-valued map )︂ , and the right-hand side : Ω → R to be specified below. We observe that the matrix-valued function satisfies (2.1) with = ∞. Further, note that depends only on the first coordinate of = ( 1 , 2 ) ∈ R 2 ; see Figure 1.

Problem with a known 0
We consider the right-hand side given by Then it is straightforward to check that the exact solution 0 ∈ 2 (Ω) ∩ 1 0 (Ω) to the homogenized problem (4.2) is given by Note that we are in the situation (Ω, , ) ∈ ℋ 2 , that = 0 at the corners of Ω and that 0 ∈ 4 (Ω). We use the scheme presented in Section 3.1 to approximate the homogenized solution 0 , where we use the same mesh for approximating and 0 . The Hsieh-Clough-Tocher (HCT) element in FreeFem++ is used in the formulation (3.6) for the 2 approximation of 0 ; see [27]. The gradient on the boundary is set to be the gradient of an 1 approximation by P 2 elements on a fine mesh.
Concerning the approximation of , from Sections 2 and 3.3 we obtain that For the numerical approximation, we replace by an 2 -conforming finite element approximation on a fine mesh, based on the formulation Find ∈ : where := 2 (Ω) ∩ 1 0 (Ω). To this end, we use again the HCT element and set the gradient on the boundary to be the gradient of an 1 approximation by P 2 elements on a fine mesh. Figure 3 shows the error in the approximation of 0 and we observe second-order convergence. Further, with the exact 0 being available, we can compute the error for different values of ; see Figure 3. We observe first-order convergence as tends to zero, as expected.

Problem with an unknown 0
Next, let us consider the problem (4.1) with the same domain Ω and matrix-valued function as before, but with the right-hand side given by Note that now we are in the situation (Ω, , ) ∈ ℋ 2 . Further, since the right-hand side ∈ 2 (Ω) of the homogenized problem (4.2) satisfies = 0 at the corners of Ω, the solution 0 to (4.2) belongs to the class 4 (Ω); see Proposition 2.6 of [29]. As before, we use the scheme presented in Section 3.1 to approximate , 0 and 0 . Using the second-order 2 (Ω)-conforming approximation 0,ℎ to 0 obtained as previously described, we have that Figure 4 shows the error ℎ 0.01 of the approximation of for different grid sizes and = 1 100 fixed. We observe fourth-order convergence in ℎ for the error as expected.
In this case we know that the homogenized problem is given by where 0 : Ω → R 2×2 is given by with being the invariant measure : ; see [22]. Therefore, we have We also note that for the corrector functions (1 ≤ , ≤ 2), i.e., the solutions to

Problem with a known 0
We consider the right-hand side given by . Then it is straightforward to check that the exact solution 0 ∈ 2 (Ω) ∩ 1 0 (Ω) to the homogenized problem (4.5) is given by Note that the assumptions of Theorem 3.15 (iii) are satisfied.
For the approximation of , Theorem 3.15 yields For the numerical approximation, we replace by an 2 -conforming finite element method on a fine mesh, based on the formulation Find ∈ : where := 2 (Ω) ∩ 1 0 (Ω). To this end, we use again the HCT element and set the gradient on the boundary to be the gradient of an 1 -conforming approximation by P 2 elements on a fine mesh.

Problem with an unknown 0
Finally, let us consider the problem (4.4) with the same domain Ω and matrix-valued function as before, but with the right-hand side given by Note that we are in the situation (Ω, , ) ∈ ℋ. Using the second-order 2 -conforming approximation 0, to 0 obtained as previously described (again with ℎ = 4 ), we have that = ( + 4 ). Figure 6 shows the error 0.02 of the approximation of for different grid sizes and = 1 50 fixed. We observe fourth-order convergence in for the error as expected.

Collection of proofs
In this section, we provide the proofs to the results presented in this paper. This section is divided into a part containing the proofs of the homogenization results from Section 2 as well as the proof of Theorem 3.15, and a second part containing the proofs of numerical results from Section 3, except for the technical Lemmata 3.6 and 3.7, which can be found in the last part of this section.

Proofs of the homogenization results
Proof of Theorem 2.5. Let us first assume that (Ω, , ) ∈ 0, for some ∈ (1, ∞). We showed in the previous section that we can transform problem (2.2) into the divergence-form problem (2.6), where div : R → R × is a -periodic, Hölder continuous, and uniformly elliptic matrix-valued function satisfying div( div ) = 0. Therefore, we can apply Theorem D from [11] to problem (2.6) to obtain with constants independent of , where we have used the property (2.4) of the invariant measure in the second inequality.
Let us now assume that (Ω, , ) ∈ ℋ 0 . Noting that (2.3) implies the Cordes condition for (︀ · )︀ with the same constant ∈ (0, 1] for any > 0, the proof of Theorem 3 from [35] yields the estimate where is the function given by We observe that by (2.1), there exist constants¯, Γ > 0 such that Therefore, we obtain from (5.1) the bound with a constant that is independent of .
Proof of Theorem 2.7. First, we note that since ∈ 0, (R ), we have ∈ 2, (R ) for any 1 ≤ , ≤ by elliptic regularity theory. A direct computation shows that the functioñ Note that since 0 ∈ 4, (Ω), one has that with the constant being independent of . We then have that :=˜− satisfies on Ω.
Therefore, by the definition of the boundary corrector, in Ω, We conclude using the estimate from Theorem 2.5 that and (2.9) holds.
Proof of Theorem 2.8. Let ∈ ∞ (R ) be a cut-off function with 0 ≤ ≤ 1, and let satisfy We introduce the function˜: and verify that where 1 , 2 and 3 are given by Therefore,˜satisfies Since 0 ∈ 4, (Ω)∩ 2,∞ (Ω) by assumption, the right-hand side belongs to (Ω), and we have by Theorem 2.5 that We look at the terms on the right-hand side separately and start with 1 . Using the boundedness of and the fact that ∈ 2,∞ (R ), we have For 2 , we obtain similarly that Finally, for 3 , we have that Altogether, we have shown that By direct computation, using the bounds we can show that Therefore, using the triangle inequality, we obtain that We conclude that The claim now follows from (2.9).

Proofs of the numerical results
Proof of Theorem 3.1. We observe that =˜+ 1, where˜is the unique solution to the problem We further observe that (3.1) is equivalent tõ We start by showing boundedness of and a Gårding-type inequality. We claim that there exist constants , > 0 such that and Let us first show (5.3). For , ∈ per ( ), by Hölder's inequality and Sobolev embeddings (note that, according to (2.1), > ), we have that Using the fact that ∈ 1, ( ) ˓→ ∞ ( ) since > , we obtain the bound for any , ∈ per ( ), i.e., (5.3) holds.
We use Schatz's method to derive an a priori estimate; see [34]. The proof of the uniqueness of˜ℎ ∈˜ℎ (which implies its existence) proceeds analogously and is therefore omitted. From our Gårding-type inequality (5.4) we see that (note that˜−˜ℎ ∈ per ( )) By Galerkin-orthogonality and boundedness, we have for any˜ℎ ∈˜ℎ that and taking the infimum over all˜ℎ ∈˜ℎ, we find that Combining this estimate with (5.5) yields Next, we use an Aubin-Nitsche-type duality argument. Let ∈ per ( ) be the unique solution to We note that the solvability condition (2.5) is satisfied: We have, using the bounds on the invariant measure (2.4), the weak formulation of (5.7) and the symmetry of , that Next, we use Galerkin orthogonality, the boundedness (5.3) and an interpolation inequality to obtain where ℐ ℎ denotes the continuous piecewise linear interpolant (for ≤ 3 and quasi-interpolant for ≥ 4) of on the triangulation. Finally, by a regularity estimate for and the bounds on the invariant measure (2.4), we arrive at the bound which provides us with the estimate ‖˜−˜ℎ‖ 2 ( ) ≤ 0 ℎ‖˜−˜ℎ‖ 1 ( ) for some 0 > 0. Combining this with (5.6) we have Therefore, for ℎ sufficiently small, we arrive at the bounds We have thus established the a priori estimate Finally, using that =˜+ 1 and ℎ =˜ℎ + 1, we conclude that Proof of Lemma 3.3. Fix 1 ≤ , ≤ . Using the definition of 0 = ( 0 ), i.e., we obtain the estimate For the first term, we have For the second term, let us first note that using ∈ 1, ( ) with > and Sobolev embeddings, we have Therefore, using a standard interpolation error bound, we obtain By Theorem 3.1, for ℎ > 0 sufficiently small, we have that Finally, we note that this implies that for ℎ > 0 sufficiently small, 0 ℎ is elliptic. Proof of Lemma 3.4. We let ℎ := 0 − ℎ 0 ∈ 2 (Ω) ∩ 1 0 (Ω) and note that ℎ is the unique solution to the boundary-value problem in Ω, We recall that 0 ∈ R × is an elliptic constant matrix. For ℎ > 0 sufficiently small, by an 2 a priori estimate, the Cauchy-Schwarz inequality and Lemma 3.3, Finally, we show that for ℎ > 0 sufficiently small, we have with the constant being independent of ℎ. This can be seen by rewriting (3.2) as with constants independent of ℎ, i.e., for ℎ > 0 sufficiently small, (5.8) holds with the constant being independent of ℎ.
holds weakly, where˜:= and denotes (︀ ·, · )︀ . Passing to the limit, we obtain that 0 is a weak solution of (3.13). We conclude the proof by noting that (3.13) admits a unique strong solution, since 0 is uniformly elliptic and Lipschitz continuous onΩ; see [25,26]. (iii) This can be proved similarly to Theorems 2.7 and 2.8, using that, by the assumptions made on and elliptic regularity, we have for any 1 ≤ , , , ≤ .

Conclusion
In this paper we introduced a scheme for the numerical approximation of elliptic problems in nondivergenceform with rapidly oscillating coefficients on 2, and polygonal domains, which is based on a 2, corrector estimate for such problems derived in the first part of this work.
We proved an optimal-order error bound for a finite element approximation of the corresponding invariant measure using continuous -periodic piecewise linear basis functions on a shape-regular triangulation of the unit cell under weak regularity assumptions on the coefficients. The coefficients are integrated against the so obtained approximation of the invariant measure after piecewise linear interpolation on the mesh to obtain an approximation of the constant coefficient-matrix of the homogenized problem. Using an 2 comparison result for the solution of this perturbed problem, we eventually obtained an approximation of the solution 0 to the homogenized problem in the 2 -norm. In the case of a polygonal domain in two space dimensions, we made use of compatibility conditions for the source term to ensure sufficiently high Sobolev-regularity of 0 .
We obtained an approximation to the solution of the original problem, i.e., the problem with oscillating coefficients, by making use of the 2 approximation of 0 , finite element approximations to second-order derivatives of the corrector functions, as well as an 2 corrector result. A method of successively approximating higher derivatives for the approximation of corrector functions was provided and analyzed. The corrector functions are necessary in order to obtain an approximation of 2 whereas the task of approximating in the 1 -norm can be achieved using only an 1 approximation of 0 .
Furthermore, we generalized our results to the case of nonuniformly oscillating coefficients, i.e., we derived an analogous corrector result and studied the approximation of the solution 0 to the homogenized problem and the solution of the -dependent problem in this case. In the final part of the paper, we presented numerical experiments matching the theoretical results for problems with both known and unknown 0 , as well as problems with nonuniformly oscillating coefficients. We illustrated the performance of the scheme for the approximation of the invariant measure, the solution 0 to the homogenized problem and the solution to the problem involving oscillating coefficients for a fixed value of .
Future work will focus on weakening of the regularity assumptions on the coefficients and the approximation of fully nonlinear nondivergence-form problems with oscillating coefficients such as the Hamilton-Jacobi-Bellman equation.