Preconditioned Iterative Methods for Diffusion Problems with High-Contrast Inclusions

TL;DR

This paper develops and analyzes robust preconditioned iterative methods for solving high-contrast diffusion problems, transforming the discretized PDE into a saddle point system and demonstrating iteration independence from contrast and mesh size.

Contribution

It introduces a novel approach converting the finite element system into a saddle point problem and proposes robust preconditioners for high-contrast diffusion scenarios.

Findings

Iterative schemes' iteration count is independent of contrast parameter.

Proposed preconditioners are effective for high-contrast inclusions.

Numerical results confirm theoretical robustness and efficiency.

Abstract

This paper concerns robust numerical treatment of an elliptic PDE with high contrast coefficients, for which classical finite-element discretizations yield ill-conditioned linear systems. 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 three preconditioned iterative procedures -- preconditioned Uzawa, preconditioned Lanczos, and PCG for the square of the matrix -- are discussed for a special type of the application, namely, highly conducting particles distributed in the domain. Robust preconditioners for solving the derived saddle point problem are proposed and investigated. Robustness with respect to the contrast parameter and the discretization scale is also justified. Numerical examples support theoretical results and demonstrate independence of the number of iterations of the proposed iterative schemes on the contrast in parameters of the problem and the mesh size.