线性波现象中拉格朗日、哈密顿与混合有限元格式的离散等价性
On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena
AI总结：
本文研究线性波传播问题中拉格朗日、辛哈密顿与混合有限元格式的离散等价性，证明在有限元相容条件下，结合蛙跳格式或隐式中点法则及梯形法则重构原始变量时，三类格式等价。
摘要：
众所周知，当场论采用变分积分器时，其拉格朗日描述与哈密顿描述在离散时间层面是等价的。除辛哈密顿结构外，许多物理系统在以混合形式书写时也表现出哈密顿结构。本文针对线性波传播现象，研究了拉格朗日格式、辛哈密顿格式与混合格式之间的离散等价性。在有限元之间满足相容性条件的情况下，拉格朗日格式与混合格式确实是等价的。在时间离散方面，考虑了蛙跳格式和隐式中点法则。在应用于波动问题的混合方法中，原始变量（例如力学中的位移或电磁学中的磁势）并不是该问题的未知量，而是根据其时间导数事后重构的。当这种重构通过梯形法则进行时，这些时间离散方法便会导致等价的格式。
英文摘要：
It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.
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