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Summary: OPTIMALITY OF MULTILEVEL PRECONDITIONERS
FOR LOCAL MESH REFINEMENT IN THREE DIMENSIONS
BURAK AKSOYLU AND MICHAEL HOLST §
Abstract. In this article, we establish optimality of the Bramble-Pasciak-Xu (BPX) norm equiv-
alence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner
in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is
to establish the optimality of BPX norm equivalence for the refinement procedures under consider-
ation. While the available optimality results for the BPX norm have been constructed primarily in
the setting of uniformly refined meshes, a notable exception is the local 2D red-green result due to
Dahmen and Kunoth. 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),
and then to use this result to extend the WHB optimality results from the quasiuniform setting to
local 2D and 3D red-green refinement scenarios.
The BPX extension is reduced to establishing that locally enriched finite element subspaces al-
low 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 refine-
ment procedure. It is possible to 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
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