Robust multigrid preconditioners for the high-contrast biharmonic plate equation. (arXiv:0910.0487v1 [math.NA])

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.

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Original source : http://arxiv.org/abs/0910.0487...