We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. When considering discontinuous Galerkin methods, one is often faced with the solution of large linear systems, especially in the case of higher-order discretisations. Trefftz discontinuous Galerkin methods allow for a reduction in the number of degrees of freedom and, thereby, the costs for solving arising linear systems significantly. In this work, we combine the concepts of geometrically unfitted finite element methods and Trefftz discontinuous Galerkin methods. From the combination of different ansatz spaces and stabilisations, we discuss a class of robust unfitted discretisations and derive a-priori error bounds, including errors arising from geometry approximation for the discretisation of a Poisson problem in a unified manner. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

翻译：本文提出了一种基于间断Trefftz试探空间的新型几何非贴体有限元方法。在考虑间断伽辽金方法时，我们常面临大型线性系统的求解难题，高阶离散情形尤为突出。Trefftz间断伽辽金方法能有效减少自由度数量，从而显著降低求解线性系统的计算成本。本研究将几何非贴体有限元方法与Trefftz间断伽辽金方法的概念相融合。通过结合不同的试探空间与稳定化策略，我们讨论了一类鲁棒的非贴体离散格式，并以统一方式推导了先验误差界（含几何近似误差），适用于泊松问题的离散求解。数值算例验证了理论分析的正确性，并展示了该方法的潜力。