In this work we propose a new, arbitrary order space-time finite element discretisation for Hamiltonian PDEs in multisymplectic formulation. We show that the new method which is obtained by using both continuous and discontinuous discretisations in space, admits a local and global conservation law of energy. We also show existence and uniqueness of solutions of the discrete equations. Further, we illustrate the error behaviour and the conservation properties of the proposed discretisation in extensive numerical experiments on the linear and nonlinear wave equation and the nonlinear Schrödinger equation.

中文翻译：

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多辛偏微分方程有限元离散化的离散守恒定律

在这项工作中，我们为多辛公式中的哈密顿偏微分方程提出了一种新的、任意阶的时空有限元离散化。我们表明，通过在空间中使用连续和不连续离散化获得的新方法，承认局部和全局能量守恒定律。我们还展示了离散方程解的存在性和唯一性。此外，我们在线性和非线性波动方程以及非线性薛定谔方程的大量数值实验中说明了所提出的离散化的误差行为和守恒性质。