Adaptive frame methods for nonlinear elliptic problems

Abstract

This article is concerned with adaptive numerical wavelet frame methods for nonlinear elliptic operator equations. The starting point of our considerations are the works of Cohen et al. [A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet schemes for nonlinear variational problems, SIAM J. Numer. Anal. 41(5) (2003), pp. 1785–1823, A. Cohen, W. Dahmen, and R. DeVore, Sparse evaluation of compositions of functions using multiscale expansions, SIAM J. Math. Anal. 35(2) (2003), pp. 279–303] in which an adaptive, asymptotically optimal evaluation scheme for certain nonlinear expressions was established using wavelet Riesz bases. Due to existing problems in the construction of such bases on domains, we adjust their ideas to the case of wavelet frames. In particular, we show that the discretization of nonlinear problems by wavelet frames, which are constructed by an overlapping partition of the domain, allows us to establish a linear convergent, adaptive algorithm for the iterative solution of the discretized problem. We then focus on main aspects of the numerical implementation of this iterative scheme. More precisely, we point out that specific non-canonical frame expansions enable us to use tree approximation ideas known from the usage of wavelet Riesz bases for the estimation of the index set of significant frame expansion coefficients of nonlinear expressions. We then prove that the ideas from Barinka [A. Barinka, Fast computation tools for adaptive wavelet schemes, Ph.D. thesis, RWTH Aachen, 2005] can be used to approximate the afore selected expansion coefficients by quadrature in linear time. Finally, concerning the complexity of the approximation, we show that all building blocks needed in the algorithm can be realized with asymptotical optimality compared with a best N-term approximation respecting tree structures, which yields an asymptotically optimal algorithm for nonlinear problems discretized by wavelet frames.