Efficient Solution of Elliptic Partial Differential Equations via Effective Combination of Mesh Quality Metrics, Preconditioners, and Sparse Linear Solvers

Abstract

In this paper, we study the effect the choice of mesh quality metric, preconditioner, and sparse linear solver have on the numerical solution of elliptic partial differential equations (PDEs). We smoothe meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and vertex condition number metrics in addition to interpolation-based, scale-variant and scale-invariant metrics. We employ the Jacobi, incomplete LU, and SSOR preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric/preconditioner/linear solver combination for the numerical solution of various elliptic PDEs.