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Consistency enforcement for the iterative solution of weak Galerkin finite element approximation of Stokes flow

Weizhang Huang, Zhuoran Wang

TL;DR

This work addresses the challenge of singular and potentially inconsistent saddle-point systems arising from the lowest-order weak Galerkin discretization of Stokes flow. By enforcing compatibility through a simple RHS modification, the authors prove that optimal-order convergence is preserved while obtaining a consistent saddle-point problem. They analyze MINRES with a block-diagonal Schur-complement preconditioner and GMRES with a block-triangular Schur-complement preconditioner, deriving eigenvalue and residual bounds that show convergence is independent of mesh size $h$ and viscosity $\mu$. Numerical experiments in 2D and 3D validate the theory, demonstrating robust, mesh- and parameter-insensitive performance of the preconditioned solvers alongside the maintained accuracy of the modified WG scheme.

Abstract

Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions.

Consistency enforcement for the iterative solution of weak Galerkin finite element approximation of Stokes flow

TL;DR

This work addresses the challenge of singular and potentially inconsistent saddle-point systems arising from the lowest-order weak Galerkin discretization of Stokes flow. By enforcing compatibility through a simple RHS modification, the authors prove that optimal-order convergence is preserved while obtaining a consistent saddle-point problem. They analyze MINRES with a block-diagonal Schur-complement preconditioner and GMRES with a block-triangular Schur-complement preconditioner, deriving eigenvalue and residual bounds that show convergence is independent of mesh size and viscosity . Numerical experiments in 2D and 3D validate the theory, demonstrating robust, mesh- and parameter-insensitive performance of the preconditioned solvers alongside the maintained accuracy of the modified WG scheme.

Abstract

Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions.

Paper Structure

This paper contains 10 sections, 88 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Weak Galerkin discretization for Stokes flow
  3. Consistency enforcement and its effects on the optimal-order convergence
  4. Convergence analysis of MINRES/GMRES with block Schur complement preconditioning
  5. Block diagonal Schur complement preconditioning
  6. Block triangular Schur complement preconditioning
  7. Numerical experiments
  8. The two-dimensional example
  9. The three-dimensional example
  10. Conclusions

Figures (1)

  • Figure 1: Examples of meshes used for the computation in two and three dimensions.

Theorems & Definitions (11)

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  • ...and 1 more