Robust multigrid preconditioners for cell-centered finite volume discretization of the high-contrast diffusion equation

TL;DR

This paper develops and analyzes a robust multigrid preconditioner for high-contrast diffusion equations discretized with cell-centered finite volume methods, demonstrating its effectiveness and independence from certain discretization choices.

Contribution

The paper extends singular perturbation analysis to cell-centered finite volume discretizations, proving the robustness of a preconditioner for high-contrast diffusion problems.

Findings

Preconditioner performance is independent of prolongation operators and smoothers.

The analysis shows the solution over highly-diffusive islands becomes constant asymptotically.

Diagonal scaling's eigenvalues depend on the number of isolated diffusive islands.

Abstract

We study a conservative 5-point cell-centered finite volume discretization of the high-contrast diffusion equation. We aim to construct preconditioners that are robust with respect to the magnitude of the coefficient contrast and the mesh size simultaneously. For that, we prove and numerically demonstrate the robustness of the preconditioner proposed by Aksoylu et al. (2008, Comput. Vis. Sci. 11, pp. 319--331) by extending the devised singular perturbation analysis from linear finite element discretization to the above discretization. The singular perturbation analysis is more involved than that of finite element because all the subblocks in the discretization matrix depend on the diffusion coefficient. However, that dependence is eliminated asymptotically. This allows the same preconditioner to be utilized due to similar limiting behaviours of the submatrices; leading to a narrowing family of preconditioners that can be used for different discretizations--a desirable preconditioner design goal. We compare our numerical results to standard cell-centered multigrid and observe that performance of our preconditioner is independent of the utilized prolongation operators and smoothers. As a side result, we also prove that the solution over the highly-diffusive island becomes constant asymptotically. Integration of this qualitative understanding of the underlying PDE to our preconditioner is the main reason behind its superior performance. Diagonal scaling is probably the most basic preconditioner for high-contrast coefficients. Extending the matrix entry based spectral analysis introduced by Graham and Hagger, we rigorously show that the number of small eigenvalues of the diagonally scaled matrix depends on the number of isolated islands comprising the highly-diffusive region.