A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems

By combining the interpolating moving least squares (IMLS) method with the variational multiscale method, a variational multiscale interpolating element-free Galerkin (VMIEFG) method is presented in this paper for the numerical solutions of convection-diffusion and Stokes problems. In the VMIEFG, the IMLS is used to construct shape functions based on using shifted and scaled polynomial bases. Compared with the variational multiscale element-free Galerkin (VMEFG) method, the VMIEFG method can directly apply the essential boundary conditions. The VMIEFG method is an effective meshless method, especially, for convection-dominated problems. Numerical examples show that the VMIEFG method avoids the oscillation in the element-free Galerkin (EFG) method, and the computational precision of the VMIEFG method is higher than that of the EFG, the VMEFG and the finite element methods.