A connection between grad-div stabilized FE solutions and pointwise divergence-free FE solutions on general meshes
SIAM Undergraduate Research Online
We prove, for Stokes, Oseen, and Boussinesq finite element discretizations on general meshes, that grad-div stabilized Taylor-Hood velocity solutions converge to the pointwise divergence-free solution (found with the iterated penalty method) at a rate of γ −1 , where γ is the grad-div parameter. However, pressure is only guaranteed to converge when (X h , ∇ · X h ) satisfies the LBB condition, where X h is the finite element velocity space. For the Boussinesq equations, the temperature solution
... emperature solution also converges at the rate γ −1 . We provide several numerical tests that verify our theory. This extends work in  which requires special macroelement structure in the mesh. Introduction It has recently been shown for several types of incompressible flow problems that on meshes with special structure that grad-div stabilized Taylor-Hood solutions converge to the Scott-Vogelius solution as the graddiv parameter approaches infinity [1, 7, 6] . Large grad-div stabilization parameters can be helpful in enforcing mass conservation in solutions, reducing error by reducing the effect of the pressure on the velocity error, and making the Schur complement easier to precondition and solve. However, a large parameter essentially enforces extra constraints on the velocity solution. Hence it is not clear if a negative effect can arise from a large parameter. By proving what the limit solution is, we can know the quality of a solution found with a large parameter. The results of our paper extend this idea to general meshes where a Scott-Vogelius solution may not exist. In such cases, convergence will be to the pointwise divergence-free solution (which is found with the iterated penalty method). In this general case, the velocity solution still converges, but the pressure solution only converges when (X h , ∇·X h ) satisfies the LBB condition, where X h is the finite element velocity space; we consider X h to be globally continuous piecewise polynomials. We consider the Stokes, Oseen, and Boussinesq systems, and for each case, we rigorously prove the convergence rates for velocity and pressure, and also temperature for Boussinesq systems. We numerically test our theory, and all computational results are consistent with our analysis. This paper is arranged as follows: Section 2 provides notation and preliminaries to allow for a smooth presentation to follow. In Section 3, we study the Stokes equations. The main result is that the grad-div stabilized Taylor-Hood velocity solutions converge to the divergence-free solution with a rate of γ −1 as γ (the grad-div parameter) approaches ∞, and a modified pressure converges to the divergence-free pressure if (X h , ∇ · X h ) satisfies the LBB condition. In Section 4, we extend the results of Section 3 to the Oseen equations and obtain analogous results. Finally, in Section 5, we further extend the results to non-isothermic Stokes flows and again find analogous results.