Adaptive wavelet methods for elliptic operator equations: Convergence rates

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution $u$ of the equation by a linear combination of $N$ wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to $u$by an arbitrary linear combination of $N$ wavelets (so called $N$-term approximation), which would be obtained by keeping the $N$largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to $u$ with error $O(N^{-s})$ in the energy norm, whenever such a rate is possible by $N$-term approximation. The range of $s>0$ for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to $N$. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.