有限元外部微积分中的多折射性

Ari Stern, Enrico Zampa
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引用次数: 0

摘要

我们考虑将有限元外部微积分(FEEC)方法应用于一类涉及微分形式的典型哈密顿 PDE 系统。这些系统的解满足局部多折射守恒律,它概括了人们更熟悉的哈密顿 ODE 系统的折射守恒律,并且与物理上重要的互易现象有关,如电磁学中的洛伦兹互易。我们描述了数值迹线满足多折射守恒律版本的混合 FEEC 方法,并将这一描述应用于几类特定的 FEEC 方法,包括保角阿诺德-福尔克-温特型方法和各种可混合的非连续加勒金(HDG)方法。有趣的是,HDG 型方法和其他非符合方法在一般意义上比符合 FEEC 方法具有更强的多折射性。这极大地扩展了麦克拉克兰和斯特恩[Found. Comput. Math., 20 (2020), pp.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Multisymplecticity in finite element exterior calculus
We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numerical traces satisfy a version of the multisymplectic conservation law, and we apply this characterization to several specific classes of FEEC methods, including conforming Arnold-Falk-Winther-type methods and various hybridizable discontinuous Galerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming methods are shown, in general, to be multisymplectic in a stronger sense than the conforming FEEC methods. This substantially generalizes previous work of McLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more restricted class of canonical Hamiltonian PDEs in the de Donder-Weyl "grad-div" form.
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