A general regularization strategy for singular Stokes problems and convergence analysis for corresponding discretization and iterative solution

TL;DR

This paper introduces a versatile regularization method for singular Stokes problems, analyzing its convergence and effectiveness with various preconditioners through theoretical proofs and numerical experiments.

Contribution

It proposes a general regularization strategy that unifies existing methods and provides convergence analysis for discretization and iterative solvers.

Findings

Optimal-order convergence is maintained with sufficient boundary data approximation.

The regularization improves the efficiency of iterative solvers like MINRES and GMRES.

Numerical experiments confirm theoretical convergence and preconditioning effectiveness.

Abstract

A general regularization strategy is considered for the efficient iterative solution of the lowest-order weak Galerkin approximation of singular Stokes problems. The strategy adds a rank-one regularization term to the zero (2,2) block of the underlying singular saddle point system. This strategy includes the existing pressure pinning and mean-zero enforcement regularization as special examples. It is shown that the numerical error maintains the optimal-order convergence provided that the nonzero Dirichlet boundary datum is approximated numerically with sufficient accuracy. Inexact block diagonal and triangular Schur complement preconditioners are considered for the regularized system. The convergence analysis for MINRES and GMRES with corresponding block preconditioners is provided for different choices of the regularization term. Numerical experiments in two and three dimensions are presented to verify the theoretical findings and the effectiveness of the preconditioning for solving the regularized system.