Summary
Two families of mixed finite elements, one based on simplices and the other on cubes, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. These spaces are analogues of those introduced by Brezzi, Douglas, and Marini in two space variables. Error estimates inL 2 andH −s are derived.
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Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing, and error estimates. M2AN,19, 7–32 (1985)
Arnold, D.N., Brezzi, F., Douglas, J., Jr.: Peers: a new mixed finite element for plane elasticity. Japan J. Appl. Math.1, 347–367 (1984)
Brezzi, F., Douglas, J., Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.47, 217–235 (1985)
Brezzi, F., Douglas, J., Jr., Marini, L.D.: Variable degree mixed methods for secod order elliptic problems. Mat. Apl. Comput.4, 19–34 (1985)
Brown, D.C.: Alternating-direction iterative schemes for mixed finite element methods for second order elliptic problems. Thesis, University of Chicago 1982
Douglas, J., Jr.: Alternating direction methods for three space variables. Numer. Math.4, 41–63 (1962)
Douglas, J., Jr., Durán, R., Pietra, P.: Formulation of alternating-direction iterative methods for mixed methods in three space. Proceedings of the Simposium Internacional de Analisis Numérico, Madrid, September 1985
Douglas, J., Jr., Durán, R., Pietra, P.: Alternating-direction iteration for mixed finite element methods. Proceedings of the Seventh International Conference on Computing Methods in Applied Sciences and Engineering, Versailles, December 1985
Douglas, J., Jr., Pietra, P.: A description of some alternating-direction iterative techniques for mixed finite element methods. Proceedings. SIAM/SEG/SPE conference, Houston, January 1985
Douglas, J., Jr., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput.44, 39–52 (1985)
Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev space. Math. Comput.34, 441–463 (1980)
Fraeijs de Veubeke, B.X.: Displacement and equilibrium models in the finite element method. In: Stress analysis (O.C. Zienkiewicz, G. Holister, eds.). New York: John Wiley 1965
Fraeijs de Veubeke, B.X.: Stress function approach. World Congress on the Finite Element Method in Structural Mechanics. Bournemouth, 1965
Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equation. Lecture Notes in Mathematics, Vol. 749. Berlin, Heidelberg, New York: Springer 1979
Nedelec, J.C.: Mixed finite elements inR 3. Numer. Math.35, 315–341 (1980)
Nedelec, J.C.: A new family of mixed finite elements inR 3 Numer. Math.50, 57–82 (1986)
Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36, 9–15 (1970/1971)
Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of the finite element method. Lecture Notes in Mathematics, Vol. 606. Berlin, Heidelberg, New York: Springer 1977
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Brezzi, F., Douglas, J., Durán, R. et al. Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987). https://doi.org/10.1007/BF01396752
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DOI: https://doi.org/10.1007/BF01396752