A conservative Galerkin finite element method for the Klein- Gordon equation in high dimensions

Abstract

School of Mathematics and Statistics, Nanjing University of Information Science & Technology, \u00a0 Nanjing, 210044, China (Received January 06 2019, accepted May 20 2019) In \u00a0this \u00a0article, \u00a0we \u00a0design \u00a0and \u00a0analyze \u00a0a \u00a0Galerkin \u00a0finite \u00a0element \u00a0method \u00a0(FEM) \u00a0to \u00a0solve \u00a0the nonlinear Klein-Gordon equation in \ud835\udc51(\ud835\udc51 = 1,2,3) dimensions. The scheme is proved to preserve well the total energy \u00a0in \u00a0the \u00a0discrete \u00a0sense, \u00a0which \u00a0is \u00a0consistent \u00a0with \u00a0the \u00a0conservative \u00a0property \u00a0possessed \u00a0by \u00a0the \u00a0original \u00a0 problem. Numerical results are reported to show the high accuracy of the numerical methods and confirm the energy conservation. \u00a0