Adaptive wavelet algorithms for elliptic PDE's on product domains

Abstract

With standard isotropic approximation by (piecewise) polynomials offixed order in a domain D subset of R-d, the convergence rate interms of the number N of degrees of freedom is inversely proportionalto the space dimension d. This so-called curse of dimensionalitycan be circumvented by applying sparse tensor product approximation,when certain high order mixed derivatives of the approximated functionhappen to be bounded in L-2. It was shown by Nitsche (2006) thatthis regularity constraint can be dramatically reduced by consideringbest N-term approximation from tensor product wavelet bases. Whenthe function is the solution of some well-posed operator equation,dimension independent approximation rates can be practically realizedin linear complexity by adaptive wavelet algorithms, assuming thatthe infinite stiffness matrix of the operator with respect to sucha basis is highly compressible. Applying piecewise smooth wavelets,we verify this compressibility for general, non-separable ellipticPDEs in tensor domains. Applications of the general theory developedinclude adaptive Galerkin discretizations of multiple scale homogenizationproblems and of anisotropic equations which are robust, i.e., independentof the scale parameters, resp. of the size of the anisotropy.