Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization

Abstract

This paper shows that use of a recently introduced sparse grad-div stabilization can increase the accuracy of projection methods for solving the Navier–Stokes equations. Sparse grad-div stabilization has recently been introduced as an alternative to standard grad-div stabilization which has a sparser matrix representation. For both sparse and standard grad-div stabilized projection methods, we prove error estimates and provide numerical experiments which reveal that both stabilizations can cause a significant decrease in the error. We then compare iterative solvers for the linear systems of equations arising from the use of both of the stabilizations. A theoretical analysis of a simplified model problem as well as numerical tests show that iterative solvers perform better for systems arising from sparse grad-div compared to standard grad-div stabilized systems.