  In Trefftz discontinuous Galerkin methods a partial differential equation is
discretized using discontinuous shape functions that are chosen to be
elementwise in the kernel of the corresponding differential operator. We
propose a new variant, the embedded Trefftz discontinuous Galerkin method,
which is the Galerkin projection of an underlying discontinuous Galerkin method
onto a subspace of Trefftz-type. The subspace can be described in a very
general way and to obtain it no Trefftz functions have to be calculated
explicitly, instead the corresponding embedding operator is constructed. In the
simplest cases the method recovers established Trefftz discontinuous Galerkin
methods. But the approach allows to conveniently extend to general cases,
including inhomogeneous sources and non-constant coefficient differential
operators. We introduce the method, discuss implementational aspects and
explore its potential on a set of standard PDE problems. Compared to standard
discontinuous Galerkin methods we observe a severe reduction of the globally
coupled unknowns in all considered cases, reducing the corresponding computing
time significantly. Moreover, for the Helmholtz problem we even observe an
improved accuracy similar to Trefftz discontinuous Galerkin methods based on
plane waves.
