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.

[1]  R. Schneider,et al.  Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme , 1995 .

[2]  Wolfgang Dahmen,et al.  Stable multiscale bases and local error estimation for elliptic problems , 1997 .

[3]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[4]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[5]  R. DeVore,et al.  Compression of wavelet decompositions , 1992 .

[6]  Ralf Kornhuber,et al.  A posteriori error estimates for elliptic problems in two and three space dimensions , 1996 .

[7]  R. Bank,et al.  Some Refinement Algorithms And Data Structures For Regular Local Mesh Refinement , 1983 .

[8]  Wolfgang Dahmen,et al.  Nonlinear Approximation and Adaptive Techniques for Solving Elliptic Operator Equations , 1997 .

[9]  Christoph Schwab,et al.  Fully Discrete Multiscale Galerkin BEM , 1997 .

[10]  Ronald R. Coifman,et al.  Multiscale Inversion of Elliptic Operators , 1998 .

[11]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[12]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[13]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[14]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[15]  Silvia Bertoluzza.  An adaptive collocation method based on interpolating wavelets , 1997 .

[16]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[17]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Elliptic Problems - Implementation and Numerical Experiments , 2001, SIAM J. Sci. Comput..

[18]  Philippe H. Tchamitchian.  Wavelets, functions, and operators , 1996 .

[19]  I. Babuska,et al.  A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator , 1987 .

[20]  R. DeVore,et al.  Interpolation spaces and non-linear approximation , 1988 .

[21]  Ronald A. DeVore,et al.  Some remarks on greedy algorithms , 1996, Adv. Comput. Math..

[22]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[23]  Silvia Bertoluzza,et al.  On the adaptive computation of integrals of wavelets , 2000 .

[24]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[25]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[26]  G. Weiss,et al.  Littlewood-Paley Theory and the Study of Function Spaces , 1991 .

[27]  Y. Meyer.  Opérateurs de Calderón-Zygmund , 1990 .

[28]  Wolfgang Dahmen,et al.  Wavelet Least Squares Methods for Boundary Value Problems , 2001, SIAM J. Numer. Anal..

[29]  Jean-Lin Journé.  Calderón-Zygmund operators , 1983 .

[30]  Silvia Bertoluzza,et al.  Wavelet stabilization of the Lagrange multiplier method , 2000, Numerische Mathematik.

[31]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[32]  Wolfgang Dahmen,et al.  Multiscale methods for pseudo-di erential equations on smooth manifolds , 1995 .

[33]  Michael T. Heath,et al.  Scientific Computing , 2018 .

[34]  Reinhard H Stephan Dahlke.  Adaptive Wavelet Methods for Saddle Point Problems , 1999 .

[35]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[36]  Wolfgang Dahmen,et al.  Multiscale Methods for Pseudo-Differential Equations on Smooth Closed Manifolds , 1994 .

[37]  Silvia Bertoluzza.  A posteriori error estimates for the wavelet Galerkin method , 1995 .

[38]  Ronald R. Coifman,et al.  Signal and image representation in combined spaces , 1998 .

[39]  James M. Keiser,et al.  An Adaptive Pseudo-Wavelet Approach for Solving Nonlinear Partial Differential Equations , 1997 .

[40]  Wolfgang Dahmen,et al.  Multigrid and Multiscale Decompositions , 1997, LSSC.

[41]  R. DeVore,et al.  Besov regularity for elliptic boundary value problems , 1997 .