Analysis of the MLS variants in the meshless local Petrov-Galerkin method for a solution to the 2D Laplace equation

We investigated the performance of various moving least squares (MLS) approximation schemes in surface fitting and in a partial-differential-equations solver based on the meshless local Petrov-Galerkin (MLPG) method. A modified weight functions in the MLS approximation and the orthogonal basis functions-based MLS ensures the Kronecker delta property. Variants of the MLS were first compared for the surface-fitting test cases. Then, the MLPG method, based on improved versions of the MLS approximations, was developed. The convergence rate, accuracy and computational time of the MLPG method were compared for the 2D Laplace equation. The MLPG method with the Kronecker delta property makes it easy to impose the essential boundary conditions, as with the finite element method (FEM), while on an orthogonal basis the function based MLS approximation increases the computational efficiency of the MLPG method, which both narrow the gap between the FEM and the MLPG.