14:00
Computer simulation of the propagation and interaction of linear waves
is a core task in computational science and engineering.
The finite element method represents one of the most common
discretisation techniques for Helmholtz and Maxwell's equations, which
model time-harmonic acoustic and electromagnetic wave scattering.
At medium and high frequencies, resolution requirements and the
so-called pollution effect entail an excessive computational effort
and prevent standard finite element schemes from an effective use.
The wave-based Trefftz methods offer a possible way to deal with this
problem: trial and test functions are special solutions of the
underlying PDE inside each element, thus the information about the
frequency is directly incorporated in the discrete spaces.
This talk is concerned with a family of those methods: the so-called
Trefftz-discontinuous Galerkin (TDG) methods, which include the
well-known ultraweak variational formulation (UWVF).
We derive a general formulation of the TDG method for Helmholtz
impedance boundary value problems and we discuss its well-posedness
and quasi-optimality.
A complete theory for the (a priori) h- and p-convergence for plane
and circular/spherical wave finite element spaces has been developed,
relying on new best approximation estimates for the considered
discrete spaces.
In the two-dimensional case, on meshes with very general element
shapes geometrically graded towards domain corners, we prove
exponential convergence of the discrete solution in terms of number of
unknowns.
This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH Zurich,
Switzerland) and Ilaria Perugia (Vienna, Austria).