Jan Glaubitz: Beyond polynomials: SBP operators for general approximation spaces

Abstract:

Solving time-dependent partial differential equations (PDEs) is vital in many fields. Preserving the PDEs' critical structures during discretization enhances the accuracy and robustness of numerical methods, especially for advection-dominated problems. Summation-by-parts (SBP) operators, which mimic integration-by-parts discretely, play a crucial role in this. They facilitate transferring stability estimates from the continuous to the discrete level.

While traditional SBP operators assume polynomial approximations of PDE solutions, there's an increasing demand for non-polynomial methods. Radial basis function (RBF) schemes and neural network-based approximations are notable examples.

In this talk, we develop a theory for function-space SBP (FSBP) operators that extend the structure-preserving benefits of polynomial SBP operators to a wider range of function spaces, allowing for more flexibility and applicability. This talk is based on joint work with Jan Nordström and Philipp Öffner.