The Mixed Finite Element Multigrid Method for Stokes Equations

Abstract

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.

Multi-grid Multi-Level Monte Carlo Method for Stokes-Darcy interface Model with Random Hydraulic Conductivity

“…Furthermore, the multi-level Monte Carlo method only reduces the computational cost in the probability space, not in the physical space. Inspired by a fact that the multi-level Monte Carlo method already has a set of hierarchical grids for the multi-level idea, it is a natural idea to fully make use of the same set of hierarchical grids to solve the discrete algebraic system by using the powerful multi-grid method [11,65,70,87,88], which can further improve the efficiency of the proposed multilevel Monte Carlo method. Meanwhile, the saved information of the multi-level Monte Carlo method on the set of hierarchical grids will also significantly reduce the computational cost of the multi-grid method.…”