On the stability of the moving least squares approximation and the element-free Galerkin method

In this paper, the stability of the moving least squares (MLS) approximation and a stabilized MLS approximation is analyzed theoretically and verified numerically. It is shown that the stability of the MLS approximation deteriorates severely as the nodal spacing decreases, while the stability of the stabilized MLS approximation is not affected by the nodal spacing. The stabilized MLS approximation is introduced into the element-free Galerkin (EFG) method to produce a stabilized EFG method. Theoretical error analysis of the stabilized EFG method is provided for boundary value problems with mixed boundary conditions of Dirichlet and Robin type. Numerical examples confirm the theoretical results, and show that the stabilized EFG method has higher computational precision and better stability than the original EFG method.