Tuesday 18 February 2020 16:00 – 17:00 A. Payatakes Seminar Room
“Adaptive Wavelet Galerkin Methods”
Dr Nikos Rekatsinas Institute of Applied and Computational Mathematics
The mathematical modelling of numerous phenomena in science, engineering and technology often leads to (systems of) Partial Differential Equations (PDEs) which typically can only be solved by numerical methods. In several applications, the solutions exhibit strong singularities caused, for instance, by non-smooth boundary parts of the domain or non-smooth forcing. In those cases, one seeks methods which are designed to be adapted to the features of interest of the solution. Our focus is on the Adaptive Wavelet Galerkin Method (awgm) for the optimal adaptive solution 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 order of the basis -being in our case a wavelet Riesz basis- in linear computational complexity. The optimality and the overall qualitative properties of the awgm depend crucially on the efficiency of the involved approximate residual evaluation scheme. An improvement of the latter is based on the reformulation of a 2nd order PDE as a well-posed first order system least squares (FOSLS) problem. As an alternative to the usual time-marching schemes, the FOSLS approach is extended to the optimal adaptive solution of simultaneous space-time variational formulations of parabolic evolutionary PDEs, better suited to efficiently approximate singularities that are local in both space and time. 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. The theoretical findings are illustrated with numerical results.