重叠网格上的时空 CutFEM I:简单连续网格运动

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Mats G. Larson, Anders Logg, Carl Lundholm
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引用次数: 0

摘要

我们提出了在两个重叠网格上计算热方程的切割有限元方法:一个静止的背景网格和一个在其内部/"顶部 "移动的重叠网格。在这里,重叠网格由简单连续运动规定,这意味着其位置随时间的函数是连续的、片断线性的。对于离散函数空间,我们使用空间连续 Galerkin 和时间不连续 Galerkin,并在两个网格之间的边界上增加了一个不连续性。有限元公式基于尼采方法,还包括两个网格之间时空边界上的积分项,该积分项模仿了标准的非连续 Galerkin 时间跳跃项。简单的连续网格运动导致时空离散化,而标准的分析方法要么失效,要么不适用。因此,我们为抛物线问题的有限元方法采用了一种似乎相对少见的能量分析框架,它具有足够的通用性和鲁棒性,适用于当前的设置。能量分析包括比标准基本估计略强的稳定性估计,以及与时间步长和网格大小相关的最优阶的先验误差估计。我们还给出了一个空间维度问题的数值结果,验证了分析误差收敛阶次。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

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来源期刊
Numerische Mathematik
Numerische Mathematik 数学-应用数学
CiteScore
4.10
自引率
4.80%
发文量
72
审稿时长
6-12 weeks
期刊介绍: Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers: 1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis) 2. Optimization and Control Theory 3. Mathematical Modeling 4. The mathematical aspects of Scientific Computing
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