Multilevel Spectral Domain Decomposition

TL;DR

This paper introduces a multilevel spectral domain decomposition method that enhances the robustness and scalability of solving large, heterogeneous elliptic PDE systems by combining spectral coarse spaces with multilevel correction strategies.

Contribution

It develops a multilevel extension of spectral domain decomposition methods with a general convergence theory, improving robustness and parallel scalability for elliptic variational problems.

Findings

The method achieves mesh-independent condition number bounds.

Numerical experiments demonstrate improved performance in 2D and 3D problems.

The approach is effective for both scalar diffusion and linear elasticity problems.

Abstract

Highly heterogeneous, anisotropic coefficients, e.g. in the simulation of carbon-fibre composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems, by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, as well as coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming, as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.