An Odyssey into Local Refinement and Multilevel Preconditioning II: Stabilizing Hierarchical Basis Methods

The concept of a stable Riesz basis plays a crucial role in the design of efficient multilevel preconditioners. In this article, we present a thorough analysis of the relationship between Riesz bases, matrix conditioning, and multilevel stability criteria, and the impact of local adaptive mesh refinement on these concepts. Wavelet-like modi cations have recently been successful in optimally stabilizing hierarchical basis methods in the setting of quasiuniform meshes. We closely examine the wavelet modi ed hierarchical basis (WMHB) methods by Vassilevski and Wang, and we extend the existing two- and three-dimensional optimality results for the quasiuniform setting to local refinement settings. Such modi cations rely primarily on establishing an optimal BPX preconditioner for the refinement procedures under consideration. The local refinement procedures we consider are the well-known red-green and red refinements in two dimensions, and their natural extensions to to three dimensions. The first article in this series established the fundamental assumption for the analysis of the WMHB preconditioner, together with a number of supporting results pertaining to BPX preconditioner. With these supporting tools in place, we prove the optimality of WMHB preconditioner. In the presence of continuously differentiable partial differential equation coefficients, we extend the optimality results for multiplicative WMHB method to locally re ned two-dimensional meshes by using two different red refinement strategies. Without such smoothness assumptions on the coefficients, we show that the early suboptimal results can also be extended to locally refined meshes. An interesting implication of the optimality of WMHB preconditioner is the H¹-stability of the linear operators used. In the limiting case where such operators reduce to L₂-projection, one can guarantee an a priori H1-stability of L₂-projection without verifying somewhat cumbersome mesh conditions after refinement has taken place. The existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satis ed. In search of optimal results, we prove that the optimality can still be achieved for the whole class of the mentioned local refinements by choosing an additive version of WMHB with coefficients in L_∞. The proof techniques allow extensions of the optimality to arbitrary spatial dimensions d ≥ 1.