Efficient Solution of Anisotropic Lattice Equations by the Recovery Method

Abstract

In a recent paper, the authors introduced the recovery method (local energy matching principle) for solving large systems of lattice equations. The idea is to construct a partial differential equation along with a finite element discretization such that the arising system of linear equations has equivalent energy as the original system of lattice equations. Since a vast variety of efficient solvers is available for solving large systems of finite element discretizations of elliptic PDEs, these solvers may serve as preconditioners for the system of lattice equations. In this paper, we will focus on both the theoretical and the numerical dependence of the method on various mesh-dependent parameters, which can be easily computed and monitored during the solution process. Systematic parameter tests have been performed which underline (a) the robustness and the efficiency of the recovery method and (b) the reliability of the control parameters, which are computed in a preprocessing step to predict the performance of the preconditioner based on the recovery method.