Energy-preserving mixed finite element methods for the Hodge wave equation

TL;DR

This paper introduces energy-preserving mixed finite element methods for the Hodge wave equation, combining finite element exterior calculus and a modified Galerkin approach to ensure exact energy conservation and optimal convergence.

Contribution

It develops a novel energy-preserving discretization framework for the Hodge wave equation using mixed finite elements and a continuous Galerkin time scheme, with proven optimal convergence.

Findings

Exact energy preservation in the discrete scheme

Optimal order convergence demonstrated

Numerical experiments confirm theoretical results

Abstract

Energy-preserving numerical methods for solving the Hodge wave equation is developed 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.