Hamiltonian structures of wave-type equations compatible with the finite element exterior calculus

Abstract

As it is widely accepted, for differential equations that reflect some physical properties it is preferable to use numerical schemes that inherit these properties. Many of such schemes are designed for Hamiltonian equations and are derived by using the Hamiltonian structures of the equations. In this paper, we formulate Hamiltonian structures for a class of wave-type equations that are compatible with the finite element exterior calculus. The finite element exterior calculus is a unified approach to designing finite element schemes for discretizing the scalar Laplacian and the vector Laplacian. In this theory, the stability result is obtained by using the Hodge theory and the Poincaré inequality. We provide Hamiltonian structures for the wave-type equations for which the schemes derived with the help of the finite element exterior calculus can be employed and thereby make combinations of structure-preserving methods and the finite element exterior calculus possible.