The reduced variational multiscale element free Galerkin method for three-dimensional steady incompressible Stokes and Navier–Stokes equations

In the paper, we extend the reduced variational multiscale element free Galerkin (RVMEFG) method to solve the three-dimensional steady incompressible Stokes and Navier–Stokes equations. This method is simplified based on the three-dimensional variational multiscale element free method (VMEFG), which the standard Galerkin discretization is used to momentum conservation equation and the variational multiscale method is used to mass conservation equation. The RVMEFG method inherits the advantages of the VMEFG method, which allows equal linear basis approximation of both velocity and pressure and avoids the Ladyzhenskaya–Babuška–Breezi (LBB) condition. Meanwhile, it can naturally generate the stabilization matrix. Furthermore, compared to the VMEFG method, the RVMEFG method can save computational cost. To verify the numerical stability, computational accuracy and efficiency of the method, five numerical problems are tested, and compared with the exact solution, VMEFG solution. It is shown that the RVMEFG method can avoid numerical oscillations and have higher computational efficiency which can save computational time. Meanwhile, it can guarantee the numerical accuracy for the three-dimensional steady incompressible Stokes and Navier–Stokes problems.