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

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.