Sparse tensor product wavelet approximation of singular functions

Monique Dauge (1) , Rob Stevenson (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Korteweg-de Vries Institute for Mathematics

On product domains, sparse-grid approximation yields optimal, dimension independent convergence rates when the function that is approximated has L^2-bounded mixed derivatives of a sufficiently high order. We show that the solution of Poisson's equation on the n-dimensional hypercube with Dirichlet boundary conditions and smooth right-hand side generally does not satisfy this condition. As suggested by P.-A. Nitsche in [Constr. Approx., 21(1) (2005), pp. 63--81], the regularity conditions can be relaxed to corresponding ones in weighted L^2 spaces when the sparse-grid approach is combined with local refinement of the set of one-dimensional wavelets indices towards the end points. In this paper, we prove that for general smooth right-hand sides, the solution of Poisson's problem satisfies these relaxed regularity conditions in any space dimension. Furthermore, since we remove log-factors from the energy-error estimates from Nitsche's work, we show that in any space dimension, locally refined sparse-grid approximation yields the optimal, dimension independent convergence rate.

Mots clés: Wavelets; sparse grids; weighted anisotropic Sobolev spaces; regularity estimates; tensor products; local refinements

Domaines: Analyse numérique [math.NA]

Monique Dauge, Rob Stevenson. Sparse tensor product wavelet approximation of singular functions. SIAM Journal on Mathematical Analysis, 2010, To appear. ⟨hal-00390752v1⟩