The paper introduces a new Meshless Local Petrov-Galerkin (MLPG) method for solving linear potential problems with high accuracy. This method, based on a local symmetric weak form (LSWF) and moving least squares (MLS) approximation, does not require a finite element mesh for interpolation or integration. The essential boundary conditions are imposed using a penalty method. The method is particularly efficient for solving Laplace and Poisson's equations, showing high rates of convergence with mesh refinement. The MLS approximation scheme is detailed, including the minimization of a weighted discrete \(L_2\) norm to determine the coefficients of the approximation. The method is flexible and can be applied to both linear and nonlinear problems, making it a promising approach for engineering applications.The paper introduces a new Meshless Local Petrov-Galerkin (MLPG) method for solving linear potential problems with high accuracy. This method, based on a local symmetric weak form (LSWF) and moving least squares (MLS) approximation, does not require a finite element mesh for interpolation or integration. The essential boundary conditions are imposed using a penalty method. The method is particularly efficient for solving Laplace and Poisson's equations, showing high rates of convergence with mesh refinement. The MLS approximation scheme is detailed, including the minimization of a weighted discrete \(L_2\) norm to determine the coefficients of the approximation. The method is flexible and can be applied to both linear and nonlinear problems, making it a promising approach for engineering applications.