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

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.