Robust multigrid preconditioners for the high-contrast biharmonic plate equation

TL;DR

This paper develops a robust multigrid preconditioner for high-contrast biharmonic plate equations that remains effective across different mesh sizes and contrast levels, extending previous methods and incorporating PDE qualitative properties.

Contribution

It introduces a preconditioner robust to contrast and mesh size for high-contrast biharmonic equations, extending prior work and including a qualitative PDE property in the design.

Findings

Preconditioner is robust against contrast and mesh size.

Numerical results confirm theoretical robustness.

Solution over high-bending islands becomes linear asymptotically.

Abstract

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.