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A two-level overlapping Schwarz method with energy-minimizing multiscale coarse basis functions

Junxian Wang, Eric Chung, Hyea Hyun Kim

TL;DR

This paper tackles solving second-order elliptic PDEs with highly oscillatory, high-contrast coefficients by developing a robust two-level overlapping Schwarz preconditioner that uses energy-minimizing constrained multiscale basis functions for the coarse space. The coarse space is constructed from local generalized eigenproblems and constrained energy minimization, with an oversampling strategy that localizes the basis and enables efficient computation. The authors provide rigorous stability analyses showing that the preconditioner’s performance is robust to coefficient contrast and subdomain overlap, and they present practical coarse spaces $V_{ms}$ whose condition-number bounds degrade only mildly with oversampling depth. Numerical experiments substantiate the theory, demonstrating strong robustness across random and uniform media and across varying overlap widths. The approach offers a scalable, efficient path to robust solvers for multiscale elliptic problems in heterogeneous media.

Abstract

A two-level overlapping Schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients, for which it is known that the standard coarse problem fails to give a robust preconditioner. In this paper, we develop energy minimizing multiscale finite element functions to form a more robust coarse problem. First, a local spectral problem is solved in each non-overlapping coarse subdomain, and dominant eigenfunctions are selected as auxiliary functions, which are crucial for the high contrast case. The required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions. Due to an exponential decay property, the minimization problem is solved locally on oversampling subdomains, that are unions of a few coarse subdomains. The coarse basis functions are therefore local and can be computed efficiently. The resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition. Numerical results are presented to validate the theory and show the performance.

A two-level overlapping Schwarz method with energy-minimizing multiscale coarse basis functions

TL;DR

This paper tackles solving second-order elliptic PDEs with highly oscillatory, high-contrast coefficients by developing a robust two-level overlapping Schwarz preconditioner that uses energy-minimizing constrained multiscale basis functions for the coarse space. The coarse space is constructed from local generalized eigenproblems and constrained energy minimization, with an oversampling strategy that localizes the basis and enables efficient computation. The authors provide rigorous stability analyses showing that the preconditioner’s performance is robust to coefficient contrast and subdomain overlap, and they present practical coarse spaces whose condition-number bounds degrade only mildly with oversampling depth. Numerical experiments substantiate the theory, demonstrating strong robustness across random and uniform media and across varying overlap widths. The approach offers a scalable, efficient path to robust solvers for multiscale elliptic problems in heterogeneous media.

Abstract

A two-level overlapping Schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients, for which it is known that the standard coarse problem fails to give a robust preconditioner. In this paper, we develop energy minimizing multiscale finite element functions to form a more robust coarse problem. First, a local spectral problem is solved in each non-overlapping coarse subdomain, and dominant eigenfunctions are selected as auxiliary functions, which are crucial for the high contrast case. The required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions. Due to an exponential decay property, the minimization problem is solved locally on oversampling subdomains, that are unions of a few coarse subdomains. The coarse basis functions are therefore local and can be computed efficiently. The resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition. Numerical results are presented to validate the theory and show the performance.

Paper Structure

This paper contains 6 sections, 9 theorems, 85 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Constrained energy minimizing multiscale basis functions
  3. Two-level overlapping Schwarz algorithm
  4. Analysis of condition numbers
  5. Analysis of condition numbers using $V_{ms}$ as a coarse space
  6. Numerical experiment

Key Result

Lemma 4.1

\newlabellemma:stable:decomposition:Vglb For any given $u$ in $V_h$, there exists $\{u_i \}_{i=0}^N$ with $u_0 \in V_{glb}$, and $u_i \in V_0(\Omega_i^\prime)$, $i\ge1$, such that and with the constant $C_0$ dependent on $\Lambda$ but independent of $\rho(x)$ and $\delta$.

Figures (1)

  • Figure 6.1: Examples of an overlapping subdomain $\Omega_i^\prime:=\Omega_{i,d,h}$ and a subregion $\widetilde{\Omega}_i:=\Omega_{i,k,H}^\delta$ containing $\Omega_{i}^\prime$.

Theorems & Definitions (14)

  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • Lemma 5.5
  • ...and 4 more