In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the “hybridized” versions of several of the most commonly used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily high-order methods on unstructured meshes.

中文翻译：

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混合不连续Galerkin方法的多凸性。

在本文中，我们证明了当将其应用到规范的哈密顿方程组的偏微分方程组时，可杂交的不连续Galerkin方法满足多重辛守恒律的充要条件。我们表明，通过几种最常用的有限元方法（包括混合，不合格和不连续的Galerkin方法）的“混合”版本，可以满足这些条件。（有趣的是，对于维数大于1的连续Galerkin方法，我们表明多重折衷性仅在较弱的意义上成立。）因此，这些通用有限元方法可用于规范哈密顿量的保结构离散化（或半离散化）。 ODE或PDE系统。