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).