Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A convergence analysis of Generalized Multiscale Finite Element Methods

Eduardo Abreu, Ciro Diaz, Juan Galvis

TL;DR

This work analyzes the convergence of Generalized Multiscale Finite Element Methods (GMsFEM) for second-order elliptic equations with heterogeneous multiscale coefficients. It builds a novel framework based on local and global eigenvector expansions and introduces norms defined by eigenvalue decay to quantify approximation quality. A coarse-space interpolation operator that combines Neumann and Dirichlet spectral components yields an a priori error bound expressed through the eigenvalues of the chosen local problems and a small modeling error under a structural assumption. The results are complemented by numerical experiments validating convergence and robustness for smooth and highly heterogeneous media, highlighting practical guidance on basis selection. Overall, the paper provides a more transparent and implementable convergence theory for GMsFEM with local spectral basis, supporting robust coarse discretizations in multiscale high-contrast settings.

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.

A convergence analysis of Generalized Multiscale Finite Element Methods

TL;DR

This work analyzes the convergence of Generalized Multiscale Finite Element Methods (GMsFEM) for second-order elliptic equations with heterogeneous multiscale coefficients. It builds a novel framework based on local and global eigenvector expansions and introduces norms defined by eigenvalue decay to quantify approximation quality. A coarse-space interpolation operator that combines Neumann and Dirichlet spectral components yields an a priori error bound expressed through the eigenvalues of the chosen local problems and a small modeling error under a structural assumption. The results are complemented by numerical experiments validating convergence and robustness for smooth and highly heterogeneous media, highlighting practical guidance on basis selection. Overall, the paper provides a more transparent and implementable convergence theory for GMsFEM with local spectral basis, supporting robust coarse discretizations in multiscale high-contrast settings.

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.

Paper Structure

This paper contains 14 sections, 13 theorems, 117 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on multiscale finite element methods
  3. Global eigenvalue problem
  4. Global Norms based on eigenvalue expansion decay
  5. Approximation using global eigenvectors
  6. Dirichlet eigenvalue problem in coarse blocks
  7. Local Neumann eigenvalue problem in coarse neighborhoods
  8. GMsFEM space construction using local eigenvalue problems
  9. A technical assumption
  10. Approximation properties of the coarse space
  11. A coarse-scale interpolation operator
  12. Interpolation approximation
  13. Numerical experiments
  14. Discussions and final comments

Key Result

Theorem 2

The operator $\mathcal{A}$ is a locally defined operator. More precisely, if $|||u|||_{2;{\Omega}}<\infty$ we have that $-\kappa^{-1}\hbox{\normalfont div}(\kappa \nabla u)$ belongs to the space $L^2({\Omega})$ and we have Moreover, we have

Figures (8)

  • Figure 1: Log-Log plots of errors in the energy norm vs. first eigenvalue out of the expansion after applying GMsFEM to Examples $1$ and $2$. (Left) source term and exact solution in (\ref{['Exemplo1']}). (Right) source term and exact solution in (\ref{['Exemplo2']}). Each color correspond to a fixed $N_D = \{ 0,5,10,15,20\}$ and ($N_D = \{ 0,1,2,3\}$ (right)) and each circle correspond to $N_N = \{1,5,10,...,40\}$.
  • Figure 2: Level plots of $3$ approximations of exact solution $u_1$ in (\ref{['Exemplo1']}). (Up-Left 2-$N_N$, 2-$N_D$), (Up-Right 3-$N_N$, 3-$N_D$) and (Down-Left 4-$N_N$, 4-$N_D$) evolving to exact solution (Down-Right).
  • Figure 3: Level plots of $3$ approximations of exact solution $u_2$ in (\ref{['Exemplo2']}). (Up-Left 1-$N_N$, 0-$N_D$), (Up-Right 2-$N_N$, 0-$N_D$) and (Down-Left 3-$N_N$, 0-$N_D$) evolving to exact solution (Down-Right).
  • Figure 4: Heterogenous high permeability $3$-channels medium in a $64 \times 64$ mesh.
  • Figure 5: A 2D layer of the $SPE10$ 2D porous medium sample from SPEProject.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Remark 1
  • Theorem 2
  • proof
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 16 more