Coupling of adaptive element free Galerkin method with variational multiscale method for two-dimensional sine-gordon equation

Abstract

In this paper, a coupling of adaptive element free Galerkin method with variational multiscale method is used to solve sine-Gordon equation in two-dimensional for the first time. Meshfree method is used where no mesh regeneration is needed comparing to finite element method. Therefore, this property facilitates the insertion of nodes which is triggered by the adaptive refinement procedure. Additional new nodes will be inserted at the high gradient regions to improve the numerical solutions. The adaptive analysis such as the refinement criteria and refinement strategy will be shown as well as the development of the modified moving least squares approximation. The performance of the proposed method is validated by solving two numerical problems. The first problem is two-dimensional large localized gradient problem with available analytical solution and the second problem is sine-Gordon equation. Numerical results proved that this method can obtain higher accuracy results compared with the conventional element free Galerkin method.