Sparse Tensor Product Wavelet Approximation of Singular Functions
Monique Dauge, Rob P. Stevenson · SIAM Journal on Mathematical Analysis · 2010
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 (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 wavelet 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.