We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time-stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider the heat equation as the partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of-lines time discretisation is enabled by an implicit extension of the bulk solution through additional stabilisation, as introduced by Lehrenfeld & Olshanskii (ESAIM: M2AN, 53:585-614, 2019). The analysis of the coupled problem relies on the Lagrange-multiplier formulation, the fact that the Lagrange-multiplier solution is equal to the normal stress at the interface and that the motion of the interface is given through rigid-body motion. This paper covers the complete stability analysis of the method and an error estimate in the energy norm. This includes the dynamic error in the domain motion resulting from the discretised ODE and the forces from the discretised PDE. To the best of our knowledge this is the first error analysis of this type of coupled moving domain problem in a fully Eulerian framework. Numerical examples illustrate the theoretical results.
翻译:我们引入了一种不合适的有限元素方法,使用Lagrange- 倍增器来研究一种 Eulelian 时间步制方法,用于将域的问题移到一个模型问题中,而域运动隐含了域运动的问题。我们认为热方程是散装域的局部差分方程(PDE),而域运动则用普通的差分方程(ODE)描述,加上通过移动界面的引力转移产生的大宗部分差分方程(ODE) 。离散是基于一个时间独立的网状网状网格上不相容的不合适的有限元素分解的。通过Lehrenfeld 和 Olshandskii (ESAIM: M2AN, 53: 585-614, 2019) 将域内局部差分方程作为部分等分方程(PDE) 。 结合问题的分析取决于Lagrange- 倍增异方方方方方方方程式的配方程式, 与界面的正常压力相等,而界面的动作是通过硬体运动运动动作来决定的。本文中通过额外的整体解部分解决办法的分解,通过额外的解法化的分解法式解决方案,通过额外的解法则涵盖该方法的完整的全体解决办法的全面分析, 的完整的全的全的理论分析, 和内部域域框架的精度分析, 包括了这个模型的精确度框架的精度分析, 的精度的精度分析。