We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

翻译：我们提出了一种基于间断Trefftz试探空间的新型几何非拟合有限元方法。Trefftz方法能够减少间断伽辽金方法中的自由度数量，从而显著降低求解线性系统的计算成本。本文证明，该方法在非拟合框架下同样能有效减少自由度。针对几何与网格间存在微小切割的情况，我们给出了一类带有不同稳定化机制的几何非拟合间断伽辽金方法的统一分析框架。通过模型泊松问题的离散化类别，我们以统一方式涵盖了稳定性分析并推导了先验误差界，包括几何近似引起的误差。本分析同样适用于Trefftz试探空间和完全多项式试探空间。数值算例验证了理论结果并展示了该方法的潜力。