Constructive Approximation

, Volume 21, Issue 1, pp 63–81

Sparse Approximation of Singularity Functions

  • Pál-Andrej Nitsche

DOI: 10.1007/s00365-004-0559-4

Cite this article as:
Nitsche, PA. Constr Approx (2004) 21: 63. doi:10.1007/s00365-004-0559-4


We are concerned with the sparse approximation of functions on the d-dimensional unit cube [0,1]d, which contain powers of distance functions to lower-dimensional k-faces (corners, edges, etc.). These functions arise, e.g., from corners, edges, etc., of domains in solutions to elliptic PDEs. Usually, they deteriorate the rate of convergence of numerical algorithms to approximate these solutions. We show that functions of this type can be approximated with respect to the H1 norm by sparse grid wavelet spaces VL, (VL) = NL, of biorthogonal spline wavelets of degree p essentially at the rate p: \[ \|u - P_Lu\|_{H^1([0,1]^d)} \leq CN_L^{-p}\,(\log_2 N_L)^s \|u\|, \qquad s = s(p,d), \] where || · || is a weighted Sobolev norm and PLu \in VL.

Sparse grids Elliptic singularities Approximation of singularities Bioorthogonal wavelets 

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  • Pál-Andrej Nitsche
    • 1
  1. 1.Seminar for Applied Mathematics, ETH-Zentrum, CH-8092 ZürichSwitzerland

Personalised recommendations