We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time-stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.
翻译:我们根据不连续的Galerkin框架,发展了不同顺序的削减有限元素方法的组合,包括具有一个和两个空间维度固定界面的双曲保护法,以及在一个空间维度上移动界面的组合。界面条件是薄弱的,以确保空间的保护和稳定性。开发了一个带有空间不连续元素的剪切FEM,并结合了对线性平流问题和静止界面声波问题的标准明确时间分步制计划。在移动界面方面,我们提议了一个基于空间和时间中线性不连续元素的时空切开FEM。我们表明,拟议的 CutFEM是保守的,能源稳定。对于固定界面来说,先验误差估计得到证明。一个和两个空间维度的数值计算支持了分析,此外,我们还表明,拟议的方法具有预期的准确性。