A Robust Preconditioner for High-Contrast Problems

TL;DR

This paper presents a new preconditioning technique for solving high-contrast PDE problems efficiently, ensuring robustness against variations in coefficients and mesh size, demonstrated through numerical experiments.

Contribution

It introduces a saddle point reformulation and a robust preconditioner for high-contrast finite element problems, improving convergence independence from contrast and mesh size.

Findings

Preconditioner is robust against contrast variations.

Number of iterations is independent of mesh size.

Numerical results confirm theoretical robustness.

Abstract

A finite-element discretization of such an equation yields a linear system whose conditioning worsens as the variations in the values of PDE coefficients becomes large. This paper introduces a procedure by which the discrete system obtained from a linear finite element discretization of the given continuum problem is converted into an equivalent linear system of the saddle point type. Then a robust preconditioner for the Lanczos method of minimized iterations for solving the derived saddle point problem is proposed. Robustness with respect to the contrast parameter and the mesh size is justified. Numerical examples support theoretical results and demonstrate independence of the number of iterations on the contrast, the mesh size and also on the different contrasts on the inclusions.