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A robust numerical method for Stokes equations based on divergence-free $H$(div) finite element methods. (English) Zbl 05770810
Summary: A computational method based on a divergence-free $H$(div) approach is presented for the Stokes equations in this article. This method is designed to find velocity approximation in an exact divergence-free subspace of the corresponding $H$(div) finite element space. That is, the continuity equation is strongly enforced a priori and the pressure is eliminated from the linear system in calculation. A strength of this approach is that the saddle-point problem for Stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available. The resulting discrete system can then be solved by using existing sophisticated solvers. The aim of this article is to demonstrate the efficiency and robustness of $H$(div) finite element methods for Stokes equations. The results not only confirm the existing theoretical results but also reveal additional advantages of the method in dealing with discontinuous boundary conditions.
##### MSC:
 65N15 Error bounds (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 76D07 Stokes and related (Oseen, etc.) flows 35B45 A priori estimates for solutions of PDE 35J50 Systems of elliptic equations, variational methods
##### Keywords:
finite element methods; divergence-free; Stokes equations