Computation of differential operators in aggregated wavelet frame coordinates

Adaptive wavelet algorithms for solving operator equations have been shown to converge with the best possible rates in linear complexity. For the latter statement, all costs are taken into account, i.e. also the cost of approximating entries from the infinite stiffness matrix with respect to the wavelet basis using suitable quadrature. A difficulty is the construction of a suitable wavelet basis on the generally non-trivially shaped domain on which the equation is posed. In view of this, recently corresponding algorithms have been proposed that require only a wavelet frame instead of a basis. By employing an overlapping decomposition of the domain, where each subdomain is the smooth parametric image of the unit cube, and by lifting a wavelet basis on this cube to each of the subdomains, the union of these collections defines such a frame. A potential bottleneck within this approach is the efficient approximation of entries corresponding to pairs of wavelets from different collections. Indeed, such wavelets are piecewise smooth with respect to mutually non-nested partitions. In this paper, considering partial differential operators and spline wavelets on the subdomains, we propose an easy implementable quadrature scheme to approximate the required entries, which allows the fully discrete adaptive frame algorithm to converge with the optimal rate in linear complexity.