Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Abstract

We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids.

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

“…can be constructed either by adaptive refinement strategies starting from the initial mesh 0 (see, e.g., [11,32]), or by some coarsening strategies starting from the final mesh L (see, e.g., [19,35]). Recently, Xu, Chen, and Nochetto [35] present a unified framework for the uniform convergence of multilevel methods for H 1 (Ω)-, H (curl, Ω)-, H (div, Ω)-multigrid method for H (curl, Ω)-elliptic problems on meshes with and without hanging nodes.…”

“…Recently, Xu, Chen, and Nochetto [35] present a unified framework for the uniform convergence of multilevel methods for H 1 (Ω)-, H (curl, Ω)-, H (div, Ω)-multigrid method for H (curl, Ω)-elliptic problems on meshes with and without hanging nodes.…”

“…In [35], Xu and Chen and Nochetto proposed a space decomposition via successive coarsening of compatible patches of l , such that…”

“…Numerical experiments in [19,35] show that their multigrid algorithms are very efficient and converge uniformly with respect to L and N h . But some assumptions on the initial mesh 0 and the fine mesh h are required to guarantee that compatible patches of l do exist (for [35]) or that l is conforming (for [19]). …”

“…The purpose of this paper is to present a unified proof for the uniform convergence of adaptive multigrid methods for (1.1)-(1.2) and (1.3)-(1.4). Here we would like to emphasize the novelty of our paper compared with [19,35] as follows:…”