A multiplicative Schwarz adaptive wavelet method for elliptic boundary value problems

Abstract

Abstract. A multiplicative Schwarz overlapping domain decomposition method is considered for solving elliptic boundary value problems. By equipping the relevant Sobolev spaces on the subdomains with wavelet bases, adaptive wavelet methods are used for approximately solving the subdomain problems. The union of the wavelet bases forms a frame for the Sobolev space on the domain as a whole. The resulting method is proven to be optimal in the sense that, in linear complexity, the iterands converge with the same rate as the sequence over N ∈ N of the best approximation from the span of the best N frame elements. Numerical results are given for the method applied to Poisson's equation.

Adaptive wavelet methods for solving operator equations: An overview

“…Numerical results reported in [SW09] show that quantitatively this multiplicative adaptive Schwarz method is much more efficient that the adaptive steepest descent method described in Sect. 6.3.…”

“…The partition of the domain into overlapping subdomains, or that of the frame into the different Riesz systems, suggest the application of a Schwarz method to solve Bu = f, being the representation of the operator equation Bu = f in frame coordinates. An multiplicative adaptive Schwarz method was studied in [SW09].…”

Adaptive wavelet frame domain decomposition methods for nonlinear elliptic equations

“…In this section, we are going to collect the tools we need to deal with the discretized equation. These building blocks appear in all kinds of adaptive wavelet algorithms as in [15, 20, 17, 22, 18, 19). Roughly speaking, it is in these methods where the adaptivity of the algorithm is established.…”

“…For our test, we fix the exact solution This is an appropriate test case for our adaptive algorithm because this function belongs to all Besov spaces $B_{\tau,\tau}^{\alpha}(\Omega),\ \alpha > 0,\ \tau^{-1} = \alpha -{1 \over 2}$ but is only contained in H α (Ω) for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\alpha < \frac{3}{2}\end{align*}\end{document}, see [19). Hence, we can expect a benefit from using an adaptive algorithm.…”

“…With additional measures to control redundancies caused by the frame approach, this Richardson‐based method was shown to converge with optimal rate. Later on to further increase efficiency, the wavelet frame idea was combined with Schwarz domain decomposition methods, see [19). This approach lead to significant improvements regarding computational time and sparsity of the solution.…”