Robust Multigrid Preconditioners for the High-Contrast Biharmonic Plate Equation

Abstract

We study the high-contrast biharmonic plate equation with Hsieh-Clough-Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11: 319-331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high-contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high-contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright (C) 2010 John Wiley & Sons, Ltd.