Robust Multilevel Methods for General Symmetric Positive Definite Operators

Abstract

A general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term "robust" refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of (nonlinear) algebraic multilevel iterations. The crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems. The method is analyzed in a rather general setting applying to the scalar elliptic equation, the equations of linear elasticity, and equations arising in the solution of Maxwell's equations. Numerical results for the scalar elliptic equation are presented showing its robust convergence behavior.1. Introduction. The discretization of many important partial differential equations (PDEs) leads to symmetric positive definite (SPD) systems. Important instances of such PDEs are the stationary heat equation, the equations of linear elasticity, and equations arising in the solution of Maxwell's equations. Since in many applications these systems can be (very) large, one is interested in designing solution schemes whose convergence rates are independent of the problem sizes, i.e., mesh parameters. Furthermore, in numerous practically important situations one deals with highly varying or otherwise degenerate coefficients accounting for the physical properties of the underlying process. Examples of such coefficients can be observed when modeling porous media flow in domains with highly varying permeability fields. Other degenerate coefficients arise, e.g., in linear elasticity when considering (even homogeneous) almost incompressible media. Overall, there are many problems for which it is desirable to construct efficient solution schemes whose convergence rates are robust with respect to the problem size as well as problem parameters.The issue of achieving robustness with respect to mesh parameters has been successfully addressed by several iterative solution schemes. Here we in particular mention domain decomposition (DD) (see, e.g., [18,24]) and multilevel/multigrid methods (see, e.g., [25,27]). Achieving robustness with respect to problem parameters has proved to be a more difficult task. For standard two-level DD methods applied to the scalar elliptic equation with varying coefficient it has been shown (see [12,24]) that robust convergence rates are obtained provided that the variations of the coefficients inside coarse grid cells are bounded.

Multiscale finite element coarse spaces for the application to linear elasticity

“…The vector-valued character and the increased kernel of the elasticity system states an important difference with scalar elliptic PDEs, which can be seen also in the approaches presented in [9,11,12,41], where generalized eigenvalue problems are solved. As mentioned in Introduction, numerical results for linear elasticity are not available yet, but a theoretical verification of the robustness of the method is already provided in [41].…”

“…As mentioned in Introduction, numerical results for linear elasticity are not available yet, but a theoretical verification of the robustness of the method is already provided in [41]. The coarse spaces constructed there will contain the rigid body modes, multiplied with a (scalar) partition of unity.…”

“…More recently, this approach has been extended in [41] from a two-level to a multilevel (AMLI, see e.g. [25,39]) method for general s.p.d.…”

“…The spectral methods in [9,11,12,41] are successfully applied to various highly heterogeneous problems in 2D even for the class of problems where multiscale and energy minimizing, see e.g. [27,35,40,42], coarse spaces fail to be fully robust.…”

Achieving Robustness Through Coarse Space Enrichment in the Two Level Schwarz Framework

“…Although the eigenvalue problems are local, can be solved in parallel and only the smallest eigenvalues are needed, this setup phase could be costly and the study of the overall cost of the algorithm is still work in progress. The related methods in [2,4] have been extended to a multilevel setting by [3,12]. Moreover, this strategy was further applied by some of the authors in the BDD and FETI frameworks [9,10].…”