Abstract:

The central idea of embedded Trefftz-DG methods is to employ discretization spaces that consist of local functions satisfying the governing PDE approximately (or exactly), thereby minimizing the number of unknowns while retaining optimal approximation properties. Trefftz-DG formulations can ben used to reduce computational cost for a variety of linear and linearized PDEs.
We formulate a general splitting of the solution space into a local part, which satisfies the local PDE within each element, and a global part, which enforces inter-element coupling and boundary conditions via standard DG formulation. We present a framework with local assumptions that is flexible enough to treat a wide class of PDEs, significantly extending the applicability of Trefftz-DG methods.
This framework allows a rigorous error analysis in this general setting. The framework is based on standard global properties of DG methods combined with suitable local properties of the PDE. We show that Trefftz-DG schemes can reduce the number of degrees of freedom without degrading asymptotic convergence rates for a variety of PDEs. We show this both through rigorous analysis and illustrate this using representative examples.