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

Weizhang Huang, Zhuoran Wang

TL;DR

Addresses efficient iterative solution of Stokes flow discretized by the lowest-order Weak Galerkin method, yielding a singular saddle-point system. Regularization by prescribing the pressure at the barycenter of the first element yields a nonsingular problem; MINRES with a block-diagonal preconditioner and GMRES with a block-triangular preconditioner are analyzed, with eigenvalue and residual bounds showing convergence factors nearly independent of $\mu$ and $h$ and iteration counts growing only logarithmically in these parameters. Numerical experiments in 2D and 3D validate the theory and demonstrate robustness and scalability of the preconditioned solvers for WG discretizations 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.

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

TL;DR

Addresses efficient iterative solution of Stokes flow discretized by the lowest-order Weak Galerkin method, yielding a singular saddle-point system. Regularization by prescribing the pressure at the barycenter of the first element yields a nonsingular problem; MINRES with a block-diagonal preconditioner and GMRES with a block-triangular preconditioner are analyzed, with eigenvalue and residual bounds showing convergence factors nearly independent of and and iteration counts growing only logarithmically in these parameters. Numerical experiments in 2D and 3D validate the theory and demonstrate robustness and scalability of the preconditioned solvers for WG discretizations 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.
Paper Structure (11 sections, 78 equations, 1 figure, 4 tables)

This paper contains 11 sections, 78 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Weak Galerkin discretization and system regularization for Stokes flow
  3. Convergence of MINRES with block diagonal Schur complement preconditioning
  4. Convergence of GMRES with block triangular Schur complement preconditioning
  5. Numerical experiments
  6. The two-dimensional example
  7. Results for $\mathcal{P}_d^{-1} \mathcal{A}$
  8. Results for $\mathcal{P}_t^{-1} \mathcal{A}$
  9. The three-dimensional example
  10. Conclusions
  11. Convergence analysis of GMRES for saddle point problems with block triangular Schur complement preconditioning

Figures (1)

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

Theorems & Definitions (8)

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