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 a parabolic 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, under an assumption on the discrete interface velocity. 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- 倍增器来研究一个 Eulerian 时间阶梯式方法,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。我们考虑在散装域中采用抛离部分差分方程(PDE),而域运动则用普通的差分方程描述(ODE),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)。离散是基于在时间独立的网格上对不合适的有限元素进行分解。线方法分解是通过Lehrenfeld 和 Olshandskii (ESAIM: M2AN, 53:585-614, 2019) 隐含部分差分异方方程(PDE) 问题的分析。Lagrange- 倍增方程的解决方案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运动运动运动运动分立,本文涵盖了方法的完整稳定性分析方法,并包含了域框架中生成的离心的离心的模型模型。 模型模型模型的模型模型模型模型分析包括了我们内部的精确的模型的精确的模型的模型。