Three-dimensional stable nonconforming parallelepiped elements for the Stokes problem.
The authors consider the system of Stokes equations: \(-\nu\Delta u+ \nabla p=f\) in \(\Omega\), \(\text{div\,}u= 0\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), where \(\Omega\in \mathbb R^3\) -- a bounded domain, \(f\) -- the given body force, \(\nu>0\) the kinematic viscosity; for simplicity, they consider \(\nu=1\). For the solution of this boundary value problem they apply the approximate method proposed by \textit{J. Douglas} et al. [Math. Model. Numer. Anal. 33, No. 4, 747--770 (1999; Zbl 0041.65115)]; it is an extension of a similar paper, published in [Appl. Math. Comput. 115, No. 2--3, 101--112 (2000; Zbl 1028.76025)] by \textit{Y. Kim} and \textit{S. Lee}, elaborated in the two-dimensional case. The plan of the paper is the following: 1) the Stokes problem is briefly reviewed in some lines at the beginning of the paper, 2) the space of finite elements approximation for the Stokes problem on parallelepiped elements is given, 3) stability results are given, using macroelement technique, 4) optimal order error estimates are derived in the final part of the paper.
stationary Stokes boundary value problem
scientific article; zbMATH DE number 1861143