2024/12/13 by Weizhang Huang, Zhuoran Wang, Huang, Weizhang +1
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Contact Mechanics and Variational Inequalities
paper · pdf · doi:10.48550/arxiv.2412.09865
Finite element discretization of Stokes problems can result in singular,\ninconsistent saddle point linear algebraic systems. This inconsistency can\ncause many iterative methods to fail to converge. In this work, we consider the\nlowest-order weak Galerkin finite element method to discretize Stokes flow\nproblems and study a consistency enforcement by modifying the right-hand side\nof the resulting linear system. It is shown that the modification of the scheme\ndoes not affect the optimal-order convergence of the numerical solution.\nMoreover, inexact block diagonal and triangular Schur complement\npreconditioners and the minimal residual method (MINRES) and the generalized\nminimal residual method (GMRES) are studied for the iterative solution of the\nmodified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES\nare established. Those bounds show that the convergence of MINRES and GMRES is\nindependent of the viscosity parameter and mesh size. The convergence of the\nmodified scheme and effectiveness of the preconditioners are verified using\nnumerical examples in two and three dimensions.\n