Consistency enforcement for the iterative solution of weak Galerkin finite element approximation of Stokes flow

TL;DR

This paper addresses the inconsistency in linear systems from weak Galerkin finite element discretization of Stokes flow, proposing a consistency enforcement method that preserves convergence and improves iterative solver performance.

Contribution

It introduces a scheme modification for weak Galerkin discretization of Stokes problems that maintains optimal convergence and enhances iterative solver robustness.

Findings

Modified scheme preserves optimal convergence.

Eigenvalue bounds show solver convergence independence from viscosity and mesh size.

Numerical tests confirm improved solver performance in 2D and 3D.

Abstract

Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions.