Unconditional stability in 3d of a sparse grad-div approximation of the Navier-Stokes equations

Inclusion of a grad-div term, $-\gamma\nabla\nabla\cdot u$, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added term $-\gamma\nabla\nabla\cdot u$ couples all velocity components, decreases sparsity and increases the condition number in the linear systems that must be solved every time step. To address these three issues various sparse grad-div regularizations and a modular grad-div method have been developed. Sparse grad-div discretizations have been previously proven to be stable in $2d$ but stability in $3d$ is unproven. Let $G^{\ast }=-diag(\partial_{x}^{2},\partial_{y}^{2},\partial_{z}^{2})$ denote the diagonal of $G=-\nabla\nabla\cdot$, and $\alpha\geq0$\ an adjustable parameter. We prove unconditional, nonlinear, long time stability in $3d$ (and $2d$) of the (space discretization suppressed) sparse grad-div method% \begin{gather*} \frac{u^{n+1}-u^{n}}{k}+u^{n}\cdot\nabla u^{n+1}+\nabla p^{n+1}-\nu\Delta u^{n+1}+\\ (\gamma+\alpha)G^{\ast}u^{n+1}-[(\gamma+\alpha)G^{\ast}-\gamma G]u^{n}% =f{\text{ .}}% \end{gather*} The discretization error of the grad-div terms is first order, comparable to the implicit method used for the other terms. We prove unconditional stability of this method for free parameter $\alpha\geq0.5\gamma$ and that the method controls the persistent size of $||\nabla\cdot u||$ in general and controls the transients in $||\nabla\cdot u||$ for a cold start. Consistent numerical tests are presented. This report also presents and proves unconditional stability of a modular, sparse grad-div method.%