 Authors
 Year
 2008
 Title
 Adaptive wavelet algorithms for elliptic PDE's on product domains
 Journal
 Mathematics of Computation
 Volume
 77
 Pages (fromto)
 7192
 ISSN
 00255718
 Issue number
 261
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain D subset of Rd, the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d. This socalled curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in L2. It was shown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best Nterm approximation from tensor product wavelet bases. When the function is the solution of some wellposed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, nonseparable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy.
 URL
 go to publisher's site
 Note
 First published in Mathematics of Computation (vol. 77 ; 261), published by the American Mathematical Society
 Permalink
 http://hdl.handle.net/11245/1.293210
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