Energy-Preserving Finite Element Methods for a Class of Nonlinear Wave Equations

Document Type

Article

Publication Date

7-3-2020

Publication Title

Applied Numerical Mathematics

Volume

157

First page number:

446

Last page number:

469

Abstract

In this paper for the first time, two kinds of energy-preserving finite element approximation schemes, which are based upon the standard finite element method (FEM) and the mixed FEM, respectively, are developed and analyzed for a class of nonlinear wave equations. The energy conservation and the optimal convergence properties are obtained for both finite element schemes in their respective norms, additionally, the energy-preserving mixed FEM can produce one-order higher approximation accuracy to the flux (the gradient of the primary unknown) in L2 norm in contrast with that of the standard FEM when the same degree piecewise polynomial is employed to construct their respective finite element spaces, which may likely result in a more accurate and more physical discrete energy conservation. Numerical experiments are carried out to validate all attained theoretical results. Furthermore, the developed energy-preserving finite element methods can be directly applied to the coupled system of nonlinear wave equations, whose energy conservation and optimal convergence properties are also confirmed by our numerical experiments.

Keywords

Nonlinear wave equations; Variable coefficients; Energy conservation; Standard finite element method (FEM); Mixed FEM; Raviart-Thomas (RT) mixed element; Optimal convergence

Disciplines

Applied Mathematics

Language

English

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