Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Mats G. Larson, Carl Lundholm
{"title":"Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution","authors":"Mats G. Larson, Carl Lundholm","doi":"10.1007/s00211-024-01413-y","DOIUrl":null,"url":null,"abstract":"<p>We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/“on top” of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are <i>discontinuous</i> and <i>piecewise constant</i>. 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. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like “shift operator” that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of 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.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"52 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerische Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00211-024-01413-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

Abstract

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/“on top” of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. 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. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like “shift operator” that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of 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.

Abstract Image

重叠网格上的时空 CutFEM II:简单不连续网格演化
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 "演化的重叠网格。在这里,重叠网格是由简单的非连续演化规定的,这意味着其位置、大小和形状随时间的函数是不连续和片断恒定的。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添加了一个不连续点。有限元计算基于尼采方法。简单的非连续网格演化产生了一种时空离散化,其空间和时间之间的结构为板状乘积结构,只需稍加修改即可应用现有的分析方法。我们沿用了 Eriksson 和 Johnson 提出的分析方法(SIAM J Numer Anal 28(1):43-77, 1991; SIAM J Numer Anal 32(3):706-740, 1995)。最大的修改是引入了类似里兹的 "移位算子",用于获得误差分析所需的离散强稳定性。移位算子将原始分析推广到某些方法中,对于这些方法,某一时刻的离散子空间并不位于随后时刻的刚度形式空间中。误差分析包括先验误差估计,该误差估计在时间步长和网格大小方面都是最佳阶次。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。
求助全文
约1分钟内获得全文 求助全文
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
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信