## The meshless local Petrov-Galerkin (MLPG) method: A simple and less-costly alternative to the finite element and boundary element methods.(English)Zbl 0996.65116

Summary: A comparison study of the efficiency and accuracy of a variety of meshless trial and test functions is presented, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are presented. In all these six MLPG methods absolutely no meshes are needed either for the interpolation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain $$\Omega_s$$ (or $$\Omega_{te})$$ centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rotes of convergence, for both the unknown variables and their derivatives.
An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the finite element method, or boundary element method. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation