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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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##### References:
 [1] R. A. Adams, Sobolev Spaces. Academic Press, New York, 1975. [2] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRO Anal. Numér.,8-R2 (1974), 120–151. [3] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. · Zbl 0383.65058 [4] M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving stationary Stokes’ equations (I). RAIRO,R-3 (1973) 33–76. · Zbl 0302.65087 · doi:10.1051/m2an/197307R300331 [5] 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, Univ. Pierre et Marie Curie, Paris, 1972. [6] V. Girault and P. A. Raviart Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Math., Springer-Verlag, Berlin, 1979. · Zbl 0413.65081 [7] P. A. Raviart and J. M. Thomas, Introduction à l’Analyse Numérique des Équations aux Derivées Partielles, Masson, Paris, 1983. [8] V. Ruas, Méthodes d’éléments finis en élasticité incompressible non linéaire et diverses contributions à l’approximation des problèmes aux limites. Thèse de Doctorat d’Etat. Univ. Pierre et Marie Curie, Paris, 1982. [9] V. Ruas, Une méthode d’éléments finis non conformes en vitesse pour le probléme de Stokes tridimensionnel. Mat. Apl. e Comp.,1 (1982), 53–73. · Zbl 0489.76049 [10] V. Ruas, A complete three-dimensional version of the quadratic velocity-constant pressure finite element method for fluid flow problems. Rev. Brasil. Comps,3, No. 2 (1983/1984), 91–97. [11] R. Stenberg, Analysis of mixed finite element methods for the Stokes problem: a unified approach. Math. Comp.,42 (1984), 9–32. · Zbl 0535.76037
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