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

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. (Comput Vis Sci 11:319–331, 2008) 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 case because all the subblocks in the discretization matrix depend on the diffusion coefficient. However, as the diffusion coefficient approaches infinity, that dependence is eliminated. 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. Therefore, we have accomplished a desirable preconditioner design goal. We compare our numerical results to standard cell-centered multigrid implementations and observe that performance of our preconditioner is independent of the utilized smoothers and prolongation operators. As a side result, we also prove a fundamental qualitative property of solution of the high-contrast diffusion equation. Namely, 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. This indicates that diagonal scaling creates a significant clustering of the spectrum, a favorable property for faster convergence of Krylov subspace solvers.