Optimality of Multilevel Preconditioners for Local Mesh Refinement in Three Dimensions

In this article, we establish optimality of the Bramble--Pasciak--Xu (BPX) norm equivalence 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 consideration. 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, and Kornhuber, and then to use this result to extend the WHB optimality results from the quasi-uniform setting to local 2D and 3D red-green refinement scenarios. The BPX 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 the shape regularity of the refined elements, but also critically on a number of geometrical properties we establish between neighboring simplices produced by the Bornemann--Erdmann--Kornhuber (BEK) refinement 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 refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H1-stability of the L2-projection. The existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied. The theoretical framework employed supports arbitrary spatial dimension $d \geq 1$ and requires no coefficient smoothness assumptions beyond those required for well-posedness in H1.