论文标题
椭圆边界价值问题的未连续的galerkin方法不固定
Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems
论文作者
论文摘要
我们提出了一种基于不连续的Trefftz Ansatz空间的新几何有限元方法。 Trefftz方法允许在不连续的Galerkin方法中降低自由度的数量，从而大大出现了解决线性系统的成本。这项工作表明，它们也是减少在不固定环境中自由度的绝佳方法。我们对具有不同稳定机制的一类几何不连续的Galerkin方法进行了统一分析，以处理几何和网格之间的小切口。我们涵盖了稳定性并得出A-Priori误差界限，包括统一的模型泊松问题的几何近似产生的误差。该分析涵盖了Trefftz和完整的多项式Ansatz空间。数值示例验证了理论发现并证明了该方法的潜力。
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.
