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Sparse approximation of singularity functions. (English) Zbl 1073.65118
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.

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
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