In this talk I propose an alternative framework for the understanding of spectral methods for evolutionary PDEs: essentially, a spectral method is simply an orthonormal basis of a separable Hilbert space, plus Galerkin conditions – everything else is (a heaving mass of) detail. The talk will be devoted to understanding some of this detail, with an emphasis on methods that respect geometric features of the underlying PDE and to their analysis in a multitude of settings.
A good spectral method should tick four boxes: stability, preservation of geometric features, rapid convergence and fast numerical algebra. We will demonstrate that there exist methods, building upon ideas from harmonic analysis and the theory of orthogonal polynomials, that attain all these goals. In particular, perhaps surprisingly, there exist geometric spectral methods that are guaranteed to satisfy all geometric features (whether invariants or inequalities) of the exact solution.
We will highlight in this talk the significant recent advances in this area, while describing areas of ongoing research and sketching open problems.
https://us05web.zoom.us/j/87562399992?pwd=UktjdnNlNDZhbUdLRUNBTzl2Q2I5Zz09