**Trefftz****-discontinuous Galerkin methods: exponential
convergence of the hp-version**

Trefftz-discontinuous
Galerkin (TDG) methods are finite element schemes in
which trial and test functions, when restricted to a mesh element, are
particular solutions of the underlying PDE.
We describe the formulation of a class of TDG methods, which includes
the well-known ultraweak variational
formulation (UWVF), for Helmholtz boundary value problems and we discuss their
well-posedness and quasi-optimality. A complete theory for the (a priori) h- and
p-convergence for plane and circular/spherical wave discrete spaces has been
developed, relying on new best approximation estimates. 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 the number
of unknowns. The dependence of the error
on the number of degrees of freedom is asymptotically better than for standard
polynomial methods, and the dependence on the frequency is taken into account
explicitly.

*This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH
Zurich, Switzerland) and Ilaria Perugia (Vienna,
Austria).*