Connection Between Grad-Div Stabilized Stokes Finite Elements and Divergence-Free Stokes Finite Elements

Abstract

In this paper, we use recently developed theories of divergence–free finite element schemes to analyze methods for the Stokes problem with grad-div stabilization. For example, we show that, if the polynomial degree is sufficiently large, the solutions of the Taylor–Hood finite element scheme converges to an optimal convergence exactly divergence–free solution as the grad-div parameter tends to infinity. In addition, we introduce and analyze a stable first-order scheme that does not exhibit locking phenomenon for large grad-div parameters.