Finite Difference/Finite Element Simulation of the Two-Dimensional Linear and Nonlinear Higgs Boson Equation in the De Sitter Space-Time

Abstract

In this work, finite element simulation of the two-dimensional linear and nonlinear form of the Higgs boson equation in de Sitter space-time is presented. The mathematical model of the problem is linear and power type nonlinear Klein-Gordon-like partial differential equations. Therefore, we discretize the temporal variable using the finite difference method and we also discretize the spatial variable using the finite element method. We use the Newton linearization technique which is one of the most useful linearization techniques for the linearization of the nonlinear partial differential equations. In the Newton method, we consider the Jacobian matrix numerically. Applying the considered numerical scheme we obtain the Bubble-like solutions in good agreement with the numerical results and theory available in the literature.