A novel multigrid method for elliptic distributed control problems

TL;DR

This paper introduces a new multigrid relaxation scheme for solving saddle-point systems from distributed control problems, demonstrating high efficiency and robustness through theoretical analysis and numerical validation.

Contribution

A novel Braess-Sarazin multigrid relaxation scheme tailored for finite element discretizations of distributed control problems with saddle-point structure.

Findings

Optimal smoothing factor derived from local Fourier analysis

Numerical experiments confirm high efficiency and robustness

Scheme performs well across different regularization parameters and grid sizes

Abstract

Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess-Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inverse of the mass matrix arising in the saddle-point system. We apply local Fourier analysis to examine the smoothing properties of the Braess-Sarazin multigrid relaxation. From our analysis, the optimal smoothing factor for Braess-Sarazin relaxation is derived. Numerical experiments validate our theoretical results. The relaxation scheme considered here shows its high efficiency and robustness with respect to the regularization parameter and grid size.