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:我们建议采用基于不连续的特雷夫茨 ansatz 空间的新的不适宜地几何限制元素方法。 在考虑不连续的加列尔金方法时,人们常常面临大型线性系统的解决方案,特别是在高阶离散的情况下。 特雷夫茨不连续的加列尔金方法可以降低自由度,从而大大降低解决新产生的线性系统的成本。 在这项工作中,我们将几何不合适的有限元素方法的概念与特雷夫茨不连续的加列尔金方法结合起来。 从不同的安萨茨空间和稳定化的结合中,我们讨论了一组强健不连续的离散,并得出了优先错误界限,包括统一地分解普瓦森问题几何近似所产生的错误。 数字实例证实了理论结论,并展示了这一方法的潜力。