The adaptive tensor product wavelet scheme: sparse matrices and the application to singularly perturbed problems

Locally supported biorthogonal wavelets are constructed on the unit interval with respect to which second-order constant coefficient differential operators are sparse. As a result, the representation of second-order differential operators on the hypercube with respect to the resulting tensor product wavelet coordinates is again sparse. The advantage of tensor product approximation is that it yields (nearly) dimension-independent rates. An adaptive tensor product wavelet method is applied to solve various singularly perturbed boundary value problems. The numerical results indicate robustness with respect to the singular perturbations. For a two-dimensional model problem this will be supported by theoretical results.