Adaptive Wavelet Methods for Elliptic Operator Equations - Convergence Rates
OSTP Journal Article
SOUTH CAROLINA UNIV COLUMBIA COLUMBIA United States
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This paper is concerned with the construction and analysis of wavelet-basedadaptive algorithms for the numerical solution of elliptic equations. These algorithmsapproximate the solution u of the equation by a linear combination of Nwavelets. Therefore, a benchmark for their performance is provided by the rate ofbest approximation to u by an arbitrary linear combination of N wavelets so calledN-term approximation, which would be obtained by keeping the N largest waveletcoefficients of the real solution which of course is unknown. The main result of thepaper is the construction of an adaptive scheme which produces an approximationto u with error ONs in the energy norm, whenever such a rate is possible byN-term approximation. The range of s 0 for which this holds is only limited bythe approximation properties of the wavelets together with their ability to compressthe elliptic operator. Moreover, it is shown that the number of arithmetic operationsneeded to compute the approximate solution stays proportional to N. Theadaptive algorithm applies to a wide class of elliptic problems and wavelet bases.The analysis in this paper puts forward new techniques for treating elliptic problemsas well as the linear systems of equations that arise from the wavelet discretization.