A convergence analysis of Generalized Multiscale Finite Element Methods

A convergence analysis of Generalized Multiscale Finite Element Methods

Eduardo Abreu, Ciro Diaz Juan Galvis Department of Applied Mathematics (IMECC)
University of Campinas (UNICAMP)
Campinas 13.083-970, SP, Brazil
Departamento de Matemáticas, Universidad Nacional de Colombia,
Carrera 45 No 26-85 - Edificio Uriel Gutierréz, Bogotá D.C. - Colombia
Abstract

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of [J. Comput. Phys. 230 (2011), 937-955]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis in view of the good scientific computing practice.

keywords:
Multiscale; GMsFEM; PDE; Elliptic.
journal: Computer Physics Communications

1 Introduction

Approximation of partial differential equations posed on domains with multiscale and heterogeneous properties appear in variety of applications. For instance, when modeling subsurface flow scenarios, subsurface properties typically vary several orders of magnitude over multiple scales. In this case, the high-contrast in the properties such as permeability raises additional issues to be consider when constructing approximation of solutions. Several multiscale models to efficiently solve flow and transport processes have been considered. In despite of many contributions, the design and mathematical analysis of high-contrast multiscale problems continue being a challenging problem; See for instance hw97 (); aarnes (); aej07 (); bo09 (); AKL (); arbogast02 (); apwy07 (); cdgw03 (); eghe05 (); jennylt03 (); cgh09 (); hughes98 (); eh09 (); ge09_1 (); ge09_1reduceddim (); sarkisguzman (); sarkisburman (); tat (). These approaches approximate the effects of the fine-scale features using a coarse mesh. They attempt to capture the fine scale effects on a coarse grid via localized basis functions. The main idea of Multiscale Finite Element Methods (MsFEMs) is to construct basis functions that are used to approximate the solution on a coarse grid. The accuracy of MsFEMs is found to be very sensitive to the particularities of the construction of the basis functions (e.g., boundary conditions of local problems). See for instance eh09 (); ehg04 (); ehw99 ()).

It is known that the construction of the basis functions need to be carefully designed in order to obtain accurate coarse-scale approximations of the solution (e.g., eh09 ()). In particular, the resulting basis functions need to have similar oscillatory behavior as the fine-scale solution. In classical multiscale methods, a number of approaches are proposed to construct basis functions, e.g., oversampling techniques or the use of limited global information (e.g., hw97 (); eh09 ()) that employs solutions in larger regions to reduce localization errors. Recently, a new and promising methodology was introduced for the construction of basis function. This methodology is referred as to Generalized Multiscale Finite Element Method (GMsFEM). The main goal of GMsFEMs is to construct coarse spaces for MsFEMs that result in accurate coarse-scale solutions. This methodology was first developed in ge09_1 (); ge09_1reduceddim (); eglw11 () in connections with the robustness of domains decomposition iterative methods for solving the elliptic equation with heterogeneous coefficients subjected to appropriate boundary conditions

(1)

where is a heterogeneous scalar field with high-contrast. In particular, it is assumed that (bounded below), while can have very large values.

A main ingredient in the construction was the use of local generalize eigenvalue problems and (possible multiscale) partition of unity functions to construct the coarse spaces. Besides using one coarse function per coarse node, in the GMsFEM it was proposed to use several multiscale basis functions per coarse node. These basis functions represent important features of the solution within a coarse-grid block and they are computed using eigenvectors of an eigenvalue problem. Then, in the works egw10 (); CEG (); EGG_MultiscaleMOR (); Review (), some studies of the coarse approximation properties of the GMsFEM were carried out. In these works and for applications to high-contrast problems, methodologies to keep small the dimension of the resulting coarse space were successfully proposed. The use of coarse spaces that somehow incorporates important modes of a (local) energy related to the problem motivated the general version of the GMsFEM. Thus,a more general and practical GMsFEM was then developed in egh12 () where several (more practical) options to compute important modes to be include in the coarse space was used. See also egt11 () for an earlier construction. It is important to mention that the methodology in egh12 () was designed for parametric and nonlinear problems and can be applied for variety of applications as it have been shown in recent developments not review here.

In this paper, we prove convergence of the GMsFEM method that uses local eigenvectors as developed in CEG (); ge09_1 (); ge09_1reduceddim (); eglw11 (). The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. Some convergence analysis of the GMsFEM, using local eigenvectors or approximation of them was obtained in egw10 (). The prove, as usual in finite element analysis, focuses on constructing interpolation operator to the coarse finite element space. The a priory error is obtained for square integrable right hand side in (2). Additionally, in egw10 () the authors make some assumptions concerning integrability of residuals and also concerning boundedness of the quotients of local energy norms with weight and where is a especial partition of unity function. These assumptions are hard to verify in practice. Moreover, in the analysis they use a Caccioppoli inequality to write energy estimates from a region to a bigger region. Therefore, extensions of the analysis in egw10 () to other equations and/or different discretization is not straightforward.

In this paper, we substantially simplify the analysis of GMsFEM methods and remove the assumptions used in egw10 () to obtain convergence, yielding a general convergence proof and more suited for computational practice. We assume square integrability of the right hand side . In order to obtain error bounds in terms of the decay of the eigenvalues used in the construction we assume that the problem is regular in the sense that the solution can be well approximated by local eigenvectors which in the case of smooth coefficients, square integrable right hand side and convex domains, is implied by the classical regularity of the problem.

It is worth to mention a main difference between the classical finite element analysis and the analysis of GMsFEM procedures for the case of heterogeneous multiscale coefficients. In the usual finite element analysis, to write the interpolation error estimates, it is assumed that the solution is smooth enough or regular enough in the classical (Sobolev) sense. This is done while using Hilbert norms (at least for elliptic problems). In the case of discontinuous multiscale coefficients, it is well know that solutions are not smooth in the classical sense. Then, the classical finite element analysis arguments do not work. In this paper, we are able to write interpolation error estimates using norms suitable for the problem at hand. In particular, to measure the “smoothness” of the solution we use the decay of the expansion of the solution in terms of global eigenvectors. This is motivated by the fact that, for a given elliptic operator, the eigenvectors are a good model for smooth functions in the scale of norms generated by powers of the operator. We define then global norms, using the decay of the expansion over global eigenvectors. We also define local norms using the decay of the expansion in terms of local eigenvectors (computed locally in a coarse node neighborhood). The main result of this paper is that we can compare the new local and global norms. With this new norms, we are able to write approximation results for the interpolation of functions that solve (2) with square integrable right hand side. We also prove error estimates in terms of the eigenvalues of the eigenvalue problem used in the construction.

The rest of the paper is organized as follow. In Section 2 we present some preliminaries on multiscale methods. In Section 3 we collect some facts on the global eigenvalue problem related to the problem. Here we introduce a scale of global norms used for the analysis. These norms measure the decay of the expansion in terms of global eigenvectors. In Sections 5 and 4 we study the local eigenvalue problems also using norms that measure the decay of the expansion in terms of the local eigenvectors. We also relate local norms to the boundary values of the eigenvalue problem. In Section 6 we review a very particular realization of the GMsFEM methodology that is the one analyzed in this paper. In Section 8 we obtain our interpolation error for the resulting method. We also write our convergence result. We present some numerical experiments in Section 9. Our numerical results verify our theoretical findings for smooth coefficients. We also consider a more practical case with heterogeneous multiscale coefficients. Finally, in Section 10 we draw some conclusions and make some final comments.

2 Preliminaries on multiscale finite element methods

In this section, we describe multiscale finite element method framework. In general terms, the MsFEMs compute the coarse-scale solution by using multiscale basis functions. It can be casted as a numerical upscaling procedure. Also as a numerical homogenization method where, instead of effective parameters representing small scale effects, basis functions are constructed that capture the small scale effects on solutions.

Multiscale techniques can be applied to variety of problems. In this paper, in order to fix ideas, we consider a second order elliptic problem with a possible multiscale high-contrast coefficient. More precisely, let (or ) be a polygonal domain. We consider the elliptic equation with heterogeneous coefficients

where is a heterogeneous scalar field with high-contrast. In particular, we assume that (bounded below), while can have very large values. We assume that and therefore might be discontinuous. The variational formulation of this problem is: Find such that

(2)

Here the bilinear form and the linear functional are defined by

and

Let be a triangulation composed by elements . We refer to the triangulation as a coarse triangulation in the sense that does not necessarily resolve all the scales in the model (in our case that would be all variations and discontinuities of ). We denote the vertices of the coarse mesh and define the neighborhood of the node by

and the neighborhood of an element by,

(3)

Using the coarse mesh we introduce coarse basis functions , where is the number of coarse basis functions. In our paper, the basis functions are supported in ; however, for , there may be multiple basis functions. MsFEMs approximate the solution on a coarse grid as , where are determined from

Once ’s are determined, one can define a fine-scale approximation of the solution by reconstructing via basis functions, .

3 Global eigenvalue problem

In this section, we recall some facts about the global eigenvalue problem associated to problem (1). We stress that the global eigenvalue problem is used in the analysis only and it is not use in the computations.

We start the presentation by introducing the global mass bilinear form. This is given by

Note that we use the coefficient in the mass matrix. The reason is that our main application in mind is on high-contrast problems and, as show in Review (); EG09 (); ge09_1 (); ge09_1reduceddim (), it is important to define the mass matrix with the coefficient . Moreover, more complicated bilinear forms can be also used as in recent developments in GMsFEM; see egh12 (); EfendievGLWESAIM12 ().

We consider the eigenvalue problem (in weak form) that seeks to find eigenfunctions and scalars such that

(4)

Denote it’s eigenvalues and eigenfunctions by and , respectively. We order eigenvalues as

(5)

We have . The eigenvalue problem (4) is the weak form of the eigenvalue problem

(6)

in with homogeneous Dirichlet boundary condition on .

We recall that the eigenvectors form a complete ()-orthonormal system of that is also orthogonal with respect to the bilinear form . Given any we can write

and compute the bilinear form as

(7)

and the bilinear form as

(8)

It is important to recall that the expansion of the solution can be explicitly given. In fact, from the weak form (2) we see that

Then we have

(9)

The eigenvector are the regular functions par excellence when working with the differential operator . In particular we stress the following fact.

Remark 1

We have that has a square integrable divergence. That is, . This follows by observing that, in a generalize sense, .

3.1 Global Norms based on eigenvalue expansion decay

In this section, we introduce a scale of norms that help measuring the decay of the expansions in terms of eigenvectors of the global eigenvalue problem. These norms are used in the a priori error estimates of our GMsFEM method. We note that, without assuming some sort of regularity of the solution of (2), it is difficult to measure the rate of the error in finite element approximations and give error estimates. For this paper, we only assume that the forcing term is square integrable in order to obtain approximation using global eigenvector. Later we consider the case of approximation using locally constructed basis functions with small support that employ local eigenvector in its design.

For any written as and , we introduce the norm defined by,

We note that these norms depend on the bilinear forms and but, in order to make notation simpler, we do not stress this dependence in our notation. Note that

In this paper, we mainly use the norm with . We have that

Then, if we can define the operator applied to by

We readily have since,

Furthermore, if we also have the following integration by parts relation, that can be verified by straightforward calculations,

(10)

We now present a characterization of using the divergence operator div and the coefficient . This implies that for even integer values of (in particular for ), the norm is computed by subassembly of similar norms in subdomains.

Theorem 2

The operator is a locally defined operator. More precisely, if we have that belongs to the space and we have

Moreover, we have


Proof. Recall that for , we have the expansion For an integer define the truncated approximation of as,

We construct the as a limit in the norm of the sequence of rescaled divergences given by . To this end, we prove that the sequence is a Cauchy sequence in the norm. Indeed, we have, by using the eigenvalue problem (6) and Remark 1, the following identity,

So that, using the orthogonality of the eigenvectors, we conclude that for every we have,

This implies the claim since the series . We conclude that there exist an function, denoted by , such that we have when . We also have that for any it holds,

which proves that .

Finally note that, by using (10), for every function we have,

 

Remark 3

Notice that if is the solution of (2) with , then, we have .

Lemma 4

Assume that and let be the solution of (2). Consider such that we have

In particular, if we have that .


Proof. Using the explicit expansion in (9), the definition of the norm and then increasing the order of eigenvalues we have

This finishes the proof.  

3.2 Approximation using global eigenvectors

In this section, we show how to obtain a priori error estimates if we use the space spanned by the first eigenvectors. Given an integer and , we define

From (5), (7), and (8) it is easy to prove the following inequality

(11)

When and we obtain the usual Friedrichs’ inequality.

If is the solution of (2) and we have the following a priori estimate.

Lemma 5

Let be the solution of (2). If if we have


Proof. Using the explicit expansion in (9), the definition of the norm and then increasing oder of eigenvalues we have

This finishes the proof.  

We observe that, in particular, we have the following a priori error estimates.

The space generated by the first eigenvalues gives good approximation spaces and the analysis becomes easy.

4 Dirichlet eigenvalue problem in coarse blocks

In this section, we study the local Dirichlet eigenvalue problem associated to problem (1). For any , we define the following bilinear forms

and

We consider the eigenvalue problems that seek eigenfunctions and scalars such that

and denote its eigenvalues and eigenvectors by and , respectively. Note that the eigenvectors form an orthonormal basis of with respect to the inner product. We order eigenvalues as

The eigenvalue problem above corresponds to the approximation of the eigenvalue problem

(12)

with homogeneous Dirichlet boundary condition on . These eigenvectors are the model of regular functions working with the differential operator . In particular, the operator is well defined and well behaved over these functions. We have that has a square integrable divergence. That is, . This follows by observing that, in a generalize sense, .

Now we use the expansion in terms of local eigenvectors and define norms based on the decay of the eigenexpansion. These local norms are the ones that naturally appear in the local interpolation errors for our interpolation operator. A main issue is to compare this local norms with the global norms defined in Section 3.1 for . For the case , we prove in Theorem 6 that the local norms can be assembled to obtain a global norms equivalent to the norm only for functions that have zero value on block boundaries.

Given any we can write

and compute the local energy bilinear form by

We can also compute the local mass bilinear as,

The local norm to measure the decay of the expansion is introduced as follows. We introduce the norm,

Note that

We consider the case . If we can define the operator by,

which is square integrable since

Additionally if , we have the following local integration by parts relation that can be verified by direct calculations,

We have the following result. This result reveals that the local norms is related to the integrability of . The proof of the following theorem follows the proof of Theorem 2 but we presented in the local setting in the interest of completeness.

Theorem 6

The operator is a locally defined operator. More precisely, if we have that belongs to the space and we have

Moreover, we have


Proof. Since we have the expansion . For any integer , truncate this expansion to get,

The sequence of rescaled divergences, , is a Cauchy sequence in the norm. Indeed, we have, by using the eigenvalue problem (12), the following identity,

So that, using the orthogonality of the eigenvectors we conclude that for every we have,

Then, there exists an function, say , such that when .

We also have that for any it holds,

which proves that .

Finally note that for every function we have,

 

Given an integer and , we define

From the analogous to (5), (7), and (8) it is easy to prove the following inequality

Lemma 7

Assume that and with . We have for ,

In particular,

5 Local Neumann eigenvalue problem in coarse neighborhoods

In this section, we study local eigenvalue problem associated to problem (1). For any , we define the following bilinear forms

and

Define if is non-empty and otherwise. We consider the eigenvalue problems that seek eigenfunctions and scalars such that

and denote its eigenvalues and eigenvectors by and , respectively. Note that the eigenvectors form an orthonormal basis of of with respect to the inner product. Note that when is empty, that is, when is a floating subdomain. We order eigenvalues as

The eigenvalue problem above corresponds to the approximation of the eigenvalue problem

(13)

with homogeneous Neumann boundary condition on and homogeneous Dirichlet boundary condition on (when non-empty).

As mentioned before when studying the global eigenvalue problem, these eigenvectors are the model of regular functions working with the differential operator . In particular the operator is well defined and well behaved over these functions. We have that has a square integrable divergence. That is, . This follows by observing that, in a generalize sense, .

Given any we can write

and compute the local energy bilinear form by

We can also compute the local mass bilinear as,

The local norm to measure the decay of the expansion is introduced as follows. We introduce the semi-norm,

Note that for we have a norm and for the semi-norms becomes a norm when restricted to non-constant functions on , more precisely,

We consider the case . If we can define the operator by,

which is square integrable since

Additionally, if , we have the following local integration by parts relation (that can be verified directly by the series expansion of both sides),

(14)

We have the following result.

Theorem 8

The operator is a locally defined operator. More precisely, if we have that belongs to the space and we have

Moreover, we have


Proof. Since we have the expansion . For any integer , truncate this expansion to get,

The sequence of rescaled divergences, , is a Cauchy sequence in the norm. Indeed, we have, by using the eigenvalue problem (13), the following identity,

So that, using the orthogonality of the eigenvectors we conclude that for every we have,

Then, there exists an function, say , such that when . We also have that for any it holds,

which proves that . Finally, note that for every function we have,

 

Remark 9

In virtue of Theorem 8 and the equality (14) we see that a necessary condition for the integrability of is that on . Note that in general, if we are not sure and we do not assume then, the integration by parts become

Therefore, doing estimates about the eigenvalue decay is harder in this case. We also mention that in the analysis presented in egw10 (), it is assume the square integrability of that, as mentioned above, implies . Assuming that the solution has zero flux across boundaries neighborhoods is not a general assumption. A main contribution of this paper is to clarify this main assumption of egw10 () and to present an analysis valid for the general case where the solution does not have null fluxes across neighborhood boundaries. As we show in our numerical experiments, for the case of a solution that is not close to a function with null flux across neighborhood boundaries, the convergence rate of the GMsFEM as introduced in egw10 () is not optimal and additional basis functions constructed from local Dirichlet eigenvalues must be introduced to recover good convergence.

Lemma 10

Assume that , and with . We have for ,

In particular,

6 GMsFEM space construction using local eigenvalue problems

In this section, we summarize the construction of coarse scale finite element spaces using a GMsFEM framework. In order to focus in the analysis of convergence we consider a particular case of the construction of spaces using the GMsFEM framework as introduced in egw10 (); EG09 (). This construction evolved to the GMsFEM method as described in egh12 (). The method presented in this paper to obtain convergence can be also carried out for the constructions in egh12 () under appropriate assumptions of the local spectral problems used for the construction of coarse spaces.

We choose the basis functions that span the eigenfunctions corresponding to small eigenvalues. We note that is a covering of . Let be a partition of unity subordinated to the covering such that and , . Define the set of coarse basis functions

(15)

where is the number of eigenvalues that will be chosen for the node ; see Babuska (); Babuska2 () for more details on the generalized finite element method using partitions of unity. Denote by , as before, the local spectral multiscale space

Define also