Sparse approximation of singularity functions. (English) Zbl 1073.65118

Nitsche, Pál-Andrej

Constructive Approximation 21, No. 1, 63-81 (2005).

The author analyzes the approximation of singular functions built from powers of distance functions to corners, edges, etc., in anisotropically refined sparse tensor product spaces. It is shown that anisotropic sparse tensor product spaces of piecewise polynomials of degree \(p\) allow approximations of corner and edge singularities in dimension 3 at a rate of \({\mathcal O}(N^{-p} (\log_2 N)^s)\) with \(s= 2p+ 3/2\) in the \(H^1\) norm and where \(N\) denotes the number of degrees of freedom in the sparse anisotropic tensor product. This estimate shows that the reduction of the convergence rate due to the higher dimension and a low Sobolev regularity can be eliminated up to a logarithmic factor. The estimation applies to any dimension \(d\) and uses sparse tensor product wavelet bases with local refinement. The result yields upper bounds on nonlinear and adaptive approximation schemes in anisotropic wavelet bases for elliptic singularities.

Reviewer: Kai Schneider (Marseille)

MSC:

65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65T60	Numerical methods for wavelets
35J25	Boundary value problems for second-order elliptic equations
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

sparse grids; wavelets; approximation; anisotropy; singular functions; sparse tensor product spaces; corner and edge singularities; convergence; Sobolev regularity; wavelet bases; elliptic singularities