Adaptive quarkonial domain decomposition methods for elliptic partial differential equations

Abstract

Abstract This paper is concerned with new discretization methods for the numerical treatment of elliptic partial differential equations. We derive an adaptive approximation scheme that is based on frames of quarkonial type, which can be interpreted as a wavelet version of $hp$ finite element dictionaries. These new frames are constructed from a finite set of functions via translation, dilation and multiplication by monomials. By using nonoverlapping domain decomposition ideas, we establish quarkonial frames on domains that can be decomposed into the union of parametric images of unit cubes. We also show that these new representation systems are stable in a certain range of Sobolev spaces. The construction is performed in such a way that, similar to the wavelet setting, the frame elements, the so-called quarklets, possess a certain number of vanishing moments. This enables us to generalize the basic building blocks of adaptive wavelet algorithms to the quarklet case. The applicability of the new approach is demonstrated by numerical experiments for the Poisson equation on $L$-shaped domains.