AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL PRECONDITIONING II: STABILIZING HIERARCHICAL BASIS METHODS

In this article, we examine the wavelet modified (or stabilized) hierarchical basis (WHB) methods of Vassilevski and Wang, and extend their original quasiuniformity-based framework and results to local 2D and 3D red-green refinement procedures. The concept of a stable Riesz basis plays a critical role in the original work on WHB, and in the design of efficient multilevel preconditioners in general. We carefully examine the impact of local mesh refinement on Riesz bases and matrix conditioning. In the analysis of WHB methods, a critical first step is to establish that the BPX preconditioner is optimal for the refinement procedures under consideration. Therefore, the first article in this series was devoted to extending the results of Dahmen and Kunoth on the optimality of BPX for 2D local red-green refinement to 3D local red-green refinement procedures introduced by Bornemann-Erdmann-Kornhuber (BEK). These results from the first article, together with the local refinement extension of the WHB analysis framework presented here, allow us to establish optimality of the WHB preconditioner on locally refined meshes in both 2D and 3D. In particular, with the minimal smoothness assumption that the PDE coefficients are in L∞, we establish optimality for the additive WHB preconditioner on locally refined 2D and 3D meshes. An interesting implication of the optimality of 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 proof techniques employed throughout the paper allow extension of the optimality results, the H1-stability of L2-projection results, and the various supporting results to arbitrary spatial dimension d ≥ 1.