A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems

TL;DR

This paper introduces a two-level iterative method for mimetic finite difference schemes solving elliptic problems, demonstrating uniform convergence and effective preconditioning regardless of mesh size.

Contribution

It presents a novel two-level algorithm that guarantees convergence and provides a uniform preconditioner for mimetic finite difference discretizations of elliptic problems.

Findings

Algorithm converges uniformly independent of mesh size

Preconditioner effectiveness is validated numerically

Numerical results support theoretical claims

Abstract

We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underling partition. We also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom. Numerical results that validate the theory are also presented.