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AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL PRECONDITIONING I: OPTIMALITY OF THE BPX
 

Summary: AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL
PRECONDITIONING I: OPTIMALITY OF THE BPX
PRECONDITIONER
BURAK AKSOYLU AND MICHAEL HOLST
Abstract. In this article, we examine the Bramble-Pasciak-Xu (BPX) preconditioner in the
setting of local 3D mesh refinement. While the available optimality results for the BPX preconditioner
have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the
2D result due to Dahmen and Kunoth, which established BPX optimality on meshes produced by a
local 2D red-green refinement. The purpose of this article is to extend this original 2D optimality
result to the local 3D red-green refinement procedure introduced by Bornemann-Erdmann-Kornhuber
(BEK). The extension is reduced to establishing that locally enriched finite element subspaces allow
for the construction of a scaled basis which is formally Riesz stable. This construction turns out to
rest not only on shape regularity of the refined elements, but also critically on a number of geometrical
properties we establish between neighboring simplices produced by the BEK refinement procedure.
We also show that the number of degrees of freedom used for smoothing is bounded by a constant
times the number of degrees of freedom introduced at that level of refinement, indicating that a
practical implementable version of the resulting BPX preconditioner for the BEK refinement setting
has provably optimal (linear) computational complexity per iteration, as well as having a uniformly
bounded condition number. The theoretical framework supports arbitrary spatial dimension d 1,
and we indicate clearly which geometrical properties established here must be re-established to show

  

Source: Aksoylu, Burak - Center for Computation and Technology & Department of Mathematics, Louisiana State University
Holst, Michael J. - Department of Mathematics, University of California at San Diego

 

Collections: Computer Technologies and Information Sciences; Mathematics