SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1433-1454, January 2021.
Optimal order error estimates of the energy-preserving numerical methods for solving the Hodge wave equation is obtained in this paper. Based on the de Rham complex, the Hodge wave equation can be formulated as a first-order system and mixed finite element methods using finite element exterior calculus is used to discretize the space. A continuous time Galerkin method, which can be viewed as a modification of the Crank--Nicolson method, is used to discretize the time which results in a full discrete method preserving the energy exactly when the source term is vanished. A projection based operator is used to establish the optimal order convergence of the proposed methods. Numerical experiments are present to support the theoretical results.

中文翻译：

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霍奇波方程的保能混合有限元方法的误差分析

SIAM数值分析学报，第59卷，第3期，第1433-1454页，2021年1月。
本文得到了求解Hodge波动方程的能量守恒数值方法的最优阶次误差估计。基于 de Rham 复数，Hodge 波动方程可以表述为一阶系统，并使用使用有限元外微积分的混合有限元方法来离散空间。连续时间伽辽金方法可以看作是 Crank--Nicolson 方法的改进，用于离散时间，这导致完全离散的方法在源项消失时精确地保留能量。基于投影的算子被用来建立所提出方法的最优阶收敛性。存在数值实验来支持理论结果。