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Piecewise solenoidal vector fields and the Stokes problem. (English) Zbl 0719.76047
A finite element approximation for the solution of the Dirichlet problem for the stationary Stokes equation describing incompressible viscous liquid in a bounded domain is proposed and investigated. In contrast to other such procedures, the vector fields of finite elements satisfy the incompressibility condition pointwise on each “triangle”. The convergence of the approximations in Sobolev spaces is established. An estimate for the error in an energy type norm is given and the optimal rate of convergence for the velocity approximation in $$L^ 2(\Omega)$$ and the pressure approximation in $$H^{-1}(\Omega)$$ is found. The proof is given in detail.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows
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