Substructuring preconditioners with novel interface solvers for general elliptic-type equations in three dimensions

TL;DR

This paper introduces two innovative substructuring preconditioners for 3D elliptic equations with discontinuous coefficients, combining simplicity and efficiency, and demonstrating near-optimal convergence in numerical experiments.

Contribution

The paper presents novel interface solvers integrated into substructuring preconditioners, enhancing robustness and ease of implementation for complex 3D elliptic problems.

Findings

Nearly optimal convergence rates achieved.

Robust performance despite large coefficient jumps.

Preconditioners are computationally inexpensive and easy to implement.

Abstract

In this paper we propose two variants of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the new preconditioners, we use the simplest coarse solver associated with the finite element space induced by the coarse partition, and construct novel interface solvers based on some new observations. The resulting preconditioners share the merits of the non-overlapping domain decomposition method (DDM) and the overlapping DDM in the sense that they not only are cheap but also are easy to implement. We apply the proposed preconditioners 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 preconditioners are nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.