 
Summary: 2009; 00:119
Robust multigrid preconditioners for the highcontrast
biharmonic plate equation
Burak Aksoylu1, Zuhal Yeter1
1 Department of Mathematics & Center for Computation and Technology, Louisiana State University
SUMMARY
We study the highcontrast 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. 319331). 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 highcontrast elliptic PDEs of order
2k, k > 2. Moreover, we prove a fundamental qualitative property of solution of the highcontrast
biharmonic plate equation. Namely, the solution over the highlybending 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.
key words: Biharmonic equation, plate equation, fourth order elliptic PDE, Schur complement, low
rank perturbation, singular perturbation analysis, highcontrast coefficients, discontinuous coefficients,
