Ruas, V. An optimal three-field finite element approximation of the Stokes system with continuous extra stresses. (English) Zbl 0797.76045 Japan J. Ind. Appl. Math. 11, No. 1, 113-130 (1994). Summary: The aim is to present a mixed finite element method for solving the two- dimensional Stokes system, written in terms of the variables velocity, pressure and extra stress tensor. A convenient functional framework is introduced, allowing simpler choices of the type of approximation for the third variable. This is assumed here to be continuous, in order to ensure its applicability to viscoelastic flow problems. In the particular second order Galerkin method that is studied, the dimension of the corresponding finite element subspace is reduced to a minimum, at least as far as local stability analyses are concerned. Cited in 9 Documents MSC: 76M10 Finite element methods applied to problems in fluid mechanics 76D07 Stokes and related (Oseen, etc.) flows 76A10 Viscoelastic fluids Keywords:mixed finite element method; stress tensor; second order Galerkin method; local stability analyses × Cite Format Result Cite Review PDF Full Text: DOI References: [1] J.H. Carneiro de Araujo, V. Ruas and M.A.M. Silva Ramos, Approximation of the three-field Stokes system via optimized quadrilateral finite elements. RAIRO Modél. Math. Anal. Numér.,27 (1993), 107–127. · Zbl 0765.76053 [2] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1976. · Zbl 0363.65083 [3] M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations (I). RAIRO Rech. Oper.,3 (1973), 33–76. · Zbl 0302.65087 [4] B. Dupire, Problemas Variacionais Lineares, sua Aproximação e Formulações Mistas. Doctoral dissertation, Departmento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, 1985. [5] M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Methods Appl. Mech. Engrg.,73 (1989), 341–350. · Zbl 0692.76002 · doi:10.1016/0045-7825(89)90073-X [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. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equations. Mathematical Aspects of Finite Element Methods in Partial Differential Equations (ed. C. de Boor), Academic Press, N.Y., 1976. [8] V. Ruas Santos, Finite element solution of 3D viscous flow problems using nonstandard degrees of freedom. Japan J. Appl. Math.,2 (1985), 415–431. · Zbl 0611.76038 · doi:10.1007/BF03167084 [9] V. Ruas, Quelques approches pour le calcul tridimensionnel en fluides visqueux incompressibles par la méthode des éléments finis. C. R. Journées NURAGIEP, Ecole Centrale de Lyon, Lyon, France, May, 1990. [10] V. Ruas, A Convergent Three-field Quadrilateral Finite Element Method for Simulating Viscoelastic Flow on Irregular Meshes. Rev. Européenne Eléments Finis,1 (1992), 391–406. · Zbl 0924.76060 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.