Robust multigrid preconditioners for the high-contrast biharmonic plate equation

We study the high-contrast biharmonic plate equation with HCT and Morley discretizations. 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. (2008, Comput. Vis. Sci. 11, pp. 319–331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretizations, 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 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 solution of 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.