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

Inclusion of a term $-γ\nabla\nabla\cdot u$, forcing $\nabla\cdot u$ to be pointwise small, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added grad-div term 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. We develop and analyze herein a synthesis of a fully decoupled, parallel sparse grad-div method of Guermond and Minev with the modular grad-div method. Let $G^{\ast}=-diag(\partial_{x}^{2},\partial_{y}^{2},\partial_{z}^{2})$ denote the diagonal of $G=-\nabla\nabla\cdot$, and $α\geq0$ an adjustable parameter. The 2-step method considered is $$\begin{eqnarray} 1 &:&\frac{\widetilde{u}^{n+1}-u^{n}}{k}+u^{n}\cdot \nabla \widetilde{u}^{n+1}+\nabla p^{n+1}-νΔ\widetilde{u}^{n+1}=f\text{ & }\nabla \cdot \widetilde{u}^{n+1}=0,\\ 2 &:&\left[ \frac{1}{k}I+(γ+α)G^{\ast }\right] u^{n+1}=\frac{1}{k }\widetilde{u}^{n+1}+\left[ (γ+α)G^{\ast }-γG\right] u^{n}. \end{eqnarray}$$ We prove its unconditional, nonlinear, long time stability in $3d$ for $α\geq0.5γ$. The analysis also establishes that the method controls the persistent size of $||\nabla\cdot u||$ in general and controls the transients in $||\nabla\cdot u||$ when $u(x,0)=0$ and $f(x,t)\neq0$ provided $α>0.5γ$. Consistent numerical tests are presented.