Application of Sobolev Gradient Method to Solve Klein Gordon Equation

Abstract

To find the minima of an energy functional, is a well known problem in physics and engineering. Sobolev gradients have proven to be affective to find the critical points of a functional. Here, we introduce a similar approach to find the solution of nonlinear Klein Gordon equation (NKGE) in a finite-element setting. The results are compared using Euclidean, weighted and unweighted Sobolev gradients. We also compare the results with Newton’s method for a test problem and show that the presented method is better than Newton’s method in this case.