Convergence analysis of iterative solution with inexact block preconditioning for weak Galerkin finite element approximation of Stokes flow

TL;DR

This paper analyzes the convergence of iterative methods for solving discretized Stokes flow equations using weak Galerkin finite elements and inexact block preconditioning, showing near viscosity-independent convergence behavior.

Contribution

It provides a theoretical convergence analysis for inexact block preconditioners applied to weak Galerkin discretizations of Stokes flow, including eigenvalue bounds and iteration estimates.

Findings

Convergence factors are nearly independent of viscosity and mesh size.

Iteration counts grow logarithmically with problem parameters.

Numerical examples confirm theoretical predictions.

Abstract

This work is concerned with the convergence of the iterative solution for the Stokes flow, discretized with the weak Galerkin finite element method and preconditioned using inexact block Schur complement preconditioning. The resulting saddle point linear system is singular and the pressure solution is not unique. The system is regularized with a commonly used strategy by specifying the pressure value at a specific location. It is analytically shown that the regularized system is nonsingular but has an eigenvalue approaching zero as the fluid kinematic viscosity tends to zero. Inexact block diagonal and triangular Schur complement preconditioners are considered with the minimal residual method (MINRES) and the generalized minimal residual method (GMRES), respectively. For both cases, the bounds are obtained for the eigenvalues of the preconditioned systems and for the residual of MINRES/GMRES. These bounds show that the convergence factor of MINRES/GMRES is almost independent of the viscosity parameter and mesh size while the number of MINRES/GMRES iterations required to reach convergence depends on the parameters only logarithmically. The theoretical findings and effectiveness of the preconditioners are verified with two- and three-dimensional numerical examples.