Sparse tensor product wavelet approximation of singular functions

Monique Dauge, Rob Stevenson

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.
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Preprint IRMAR  09-23 (May 28, 2009).

SIAM Journal on Mathematical Analysis 42 No. 5 (2010) 2203-2228. DOI: 10.1137/090764694

Reprint from SIAM Journal on Mathematical Analysis (367 k)