A general regularization strategy for singular Stokes problems and convergence analysis for corresponding discretization and iterative solution
Weizhang Huang, Zhuoran Wang
TL;DR
This work addresses solving singular Stokes problems discretized by the lowest-order weak Galerkin method by introducing a general rank-one regularization term $- rac{\rho}{\mu}\mathbf{w}\mathbf{w}^T$ added to the zero $(2,2)$ block. The authors prove the regularized system is nonsingular and preserves optimal convergence provided the nonzero Dirichlet data is approximated with sufficient accuracy, and they develop MINRES and GMRES analyses with block Schur-complement preconditioners for both finite $\gamma,\rho$ and small $\gamma,\rho$ regimes. Key theoretical results include eigenvalue bounds for the approximate Schur complement, residual estimates for the preconditioned MINRES and GMRES, and robustness with respect to $\mu$ and mesh size $h$. Numerical experiments in 2D and 3D validate the theory and demonstrate the preconditioners’ effectiveness across different regularization choices, confirming practical impact for efficiently solving singular Stokes systems.
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.