Numerical Analysis and Computation of a Subgrid-Sparse-Grad-Div Stabilization Method for Incompressible Flow Problems

Abstract

In this article, a subgrid-sparse-grad-div method for incompressible flow problem was proposed, which is a combination of the subgrid stabilization method and the recently proposed sparse-grad-div method. The method maintains the advantage of both methods: (i) It is robust for solving incompressible flow problem with dominance of the convection, especially when the viscosity is too small. (ii) It can keep mass conservation. Therefore, the method is very efficient for solving incompressible flow. Moreover, based on the Crank–Nicolson extrapolated scheme for temporal discretization, and mixed finite element in spatial discretization, we derive the unconditional stability and optimal convergence of the method. Finally, numerical experiments are proposed to validate the theoretical predictions and demonstrate the efficiency of the method on a test problem for incompressible flow.