SC Seminar Nikos Rekatsinas (UvA)

Adaptive Wavelet Methods for solving First Order System Least Squares

We discuss Adaptive Wavelet Galerkin Methods (awgm) for the optimal adaptive solution of variational formulations of stationary, and evolutionary PDEs. Adaptive approximation allows the local resolution of the approximation space to be adjusted to the local smoothness of the solution. Optimality of the solution means it can be approximated at the best possible rate allowed by the basis -being in our case a wavelet Riesz basis- in optimal computational complexity. At first we show that any well-posed 2nd order PDE can be reformulated as a well-posed first order system least squares (FOSLS) problem, which we subsequently solve using the awgm. On this ground we design a novel, more efficient approximate residual evaluation scheme that improves the overall quantitative properties of the awgm. As an alternative to the usual time-marching schemes, which are not suited to efficiently approximate singularities that are local in both space and time, we extend our approach to the optimal adaptive solution of simultaneous space-time variational formulations of parabolic evolutionary PDEs. The use of tensor products of temporal and spatial wavelets allows for the whole time evolution problem to be solved at a complexity of solving one instance of the corresponding stationary problem. We illustrate our findings with numerical results.