Multigrid Methods for Saddle Point Problems: Darcy Systems

TL;DR

This paper develops and analyzes multigrid algorithms for saddle point problems from mixed finite element discretizations of Darcy systems, proving uniform convergence and extending to general elliptic problems.

Contribution

It introduces multigrid methods with proven uniform convergence for saddle point Darcy problems, including extensions to broader elliptic equations.

Findings

Proved uniform convergence of the W-cycle multigrid algorithm.

Established effectiveness for Raviart-Thomas-Nédélec mixed finite element discretizations.

Extended methods to general second order elliptic problems.

Abstract

We design and analyze multigrid methods for the saddle point problems resulting from Raviart-Thomas-N\'ed\'elec mixed finite element methods (of order at least 1) for the Darcy system in porous media flow. Uniform convergence of the $W$-cycle algorithm in a nonstandard energy norm is established. Extensions to general second order elliptic problems are also addressed.