Direct meshless local Petrov–Galerkin method for the two-dimensional Klein–Gordon equation

In this paper we apply the direct meshless local Petrov–Galerkin (DMLPG) method to solve the two dimensional Klein–Gordon equations in both strong and weak forms. Low computational cost is the main property of this method compared with the original MLPG technique. The reason lies behind the approach of generalized moving least squares approximation where the discretized functionals, obtained from the PDE problem, are directly approximated from nodal values. This shifts the integration over polynomials rather than the MLS shape functions, leading to an extremely faster scheme. We will see that this method can successfully solve the problem with a reasonable accuracy.