Sparse tensor product wavelet approximation of singular functions
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.
Preprint IRMAR 09-23 (May 28, 2009).
SIAM Journal on Mathematical Analysis 42 No. 5 (2010) 2203-2228.