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We propose a new geometrically unfitted finite element method based ondiscontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction inthe number of degrees of freedom in discontinuous Galerkin methods, thereby,the costs for solving arising linear systems significantly. This work showsthat they are also an excellent way to reduce the number of degrees of freedomin an unfitted setting. We present a unified analysis of a class ofgeometrically unfitted discontinuous Galerkin methods with differentstabilisation mechanisms to deal with small cuts between the geometry and themesh. We cover stability and derive a-priori error bounds, including errorsarising from geometry approximation for the class of discretisations for amodel Poisson problem in a unified manner. The analysis covers Trefftz and fullpolynomial ansatz spaces, alike. Numerical examples validate the theoreticalfindings and demonstrate the potential of the approach.

Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems