 
Summary: NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Numer. Linear Algebra Appl. 2009; 00:119 Prepared using nlaauth.cls [Version: 2002/09/18 v1.02]
Robust multigrid preconditioners for the highcontrast
biharmonic plate equation
Burak Aksoylu1,2, Zuhal Yeter2
1 TOBB University of Economics and Technology, Department of Mathematics, Ankara, 06560, Turkey
2 Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803, USA
SUMMARY
We study the highcontrast biharmonic plate equation with HCT 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. (2008, Comput. Vis. Sci. 11, pp. 319331). 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 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. Copyright c 2009 John
