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Erschienen in:
27.11.2015
Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition
Erschienen in: Journal of Scientific Computing | Ausgabe 1/2016
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Abstract
We consider the penalty method for the stationary Navier–Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate \(O(\epsilon )\) in \(H^k\)-norm, where \(\epsilon \) is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate \(O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })\) for the non-reduced-integration scheme with \(d=2,3\), and the reduced-integration scheme with \(d=3\), where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with \(d=2\), we prove the convergence order \(O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })\). The theoretical results are verified by numerical experiments.