A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions

TL;DR

This paper introduces a new substructuring preconditioner for 3D elliptic equations with discontinuous coefficients, featuring vertex-related interface solvers that are simple, efficient, and independent of the specific equation form.

Contribution

It presents a novel preconditioner using vertex-related interface solvers and a simple coarse solver, effective for various elliptic-type equations with discontinuous coefficients.

Findings

Nearly optimal convergence rate in PCG method

Robustness against large coefficient jumps

Easy to implement and computationally cheap

Abstract

In this paper we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition, and construct vertex-related inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. % in the sense that they not only are cheap but also are easy to implement. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell's equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.