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Finite element solution of 3-D viscous flow problems using nonstandard degrees of freedom. (English) Zbl 0611.76038
The author presents a detailed study of well-posedness and convergence of two-mixed finite element methods for solving three-dimensional incompressible Stokes equations in a polyhedral convex domain \(\Omega\) of \({\mathbb{R}}^ 3\). They are generalizations to the three-dimensional case of two classical velocity-pressure triangular elements: the (P2)-(P0) finite element of M. Fortin [Calcul numérique des écoulements des fluides de Bingham et des fluides Newtoniens incompressibles par la méthode des éléments finis, Thèse de Doctorat d’Etat, Université Pierre-et-Marie-Curie, Paris (1972)] and its ”bubble- version” that was proposed by M. Crouzeix and P. A. Raviart [Revue Franc. Automat. Inform. Rech. Operat. 7(1973), R3, 33-76 (1974; Zbl 0302.65087)]. For each h, the domain \(\Omega\) is split into a finite number of tetrahedrons with maximal edge length of the order of h. The family \(\{\tau_ h\}\) of three-dimensional triangulations is assumed to be regular in the sense that there exists a lower bound \(\theta_ 0>0\) for the angles between adjacent faces of the tetrahedrons of \(\tau_ h\), for all h.
For both finite elements the author proves optimal convergence results. More precisely, for the finite element i, \(i=1,2\), assuming that the velocity belongs to \(H^{1+i}(\Omega)^ 3\) and that the pressure belongs to \(H^ i(\Omega)\), he obtains an \(O(h^ i)\) error estimate for the \(H^ 1(\Omega)^ 3\)-norm of the velocity and for the \(L^ 2(\Omega)\)-norm of the pressure. To establish these convergence results, the author proves in passing that a discrete Babuška-Brezzi inf-sup condition holds for these finite elements. Next, he uses standard approximation results (Cea’s lemma) to conclude.
Reviewer: C.Conca

MSC:
76D07 Stokes and related (Oseen, etc.) flows
65N15 Error bounds for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
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