Numer. Math. 51,237-250 (1987) Numerische Mathematik 9 Springer-Verlag 1987 Mixed Finite Elements for Second Order Elliptic Problems in Three Variables Franco Brezzi 1, Jim Douglas, Jr. 2, Ricardo DurUm 2, and Michel Fortin 3 Dipartimento di Meccanica Strutturale, Universith di Pavia, and Istituto di Analisi Numerica del C.N.R. di Pavia, 1-27100 Pavia, Italy 2 Department of Mathematics, University of Chicago, Chicago, IL 60637, USA 3 D~partement de Math6matiques, Universit6 Laval, Quebec G1K 7P4, Canada 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 in L2 and H -s are derived. Subject Classifications: AMS(MOS): 65N30; CR: G1.8. 1. Introduction We introduce two families of spaces of mixed finite elements to approximate the solutions of second order elliptic equations in three space variables. These families are the reasonable analogues of the spaces recently described by Brezzi et al. [3, 4] for two-dimensional problems. For the simplicial elements our space of index j lies between the spaces of index j-1 and j of Nedelec [15] and provides approximation of the vector variable of the same order of accuracy as does Nedelec's space of index j. Our cubic elements of index j are based on polynomials of total degree j for the vector variable and total degree j-1 for the scalar variable; hence, the local dimension of this space is about half that of the Raviart-Thomas [18] space over cubes (i.e., rectangular par- allelepipeds) of equivalent accuracy for the vector variable. Nedelec [16] has recently considered the same tetrahedral spaces. In Sect. 2 we describe the simplicial elements and locally defined pro- jections that enable us to use the theory of Douglas and Roberts [10] to obtain error estimates for the Dirichlet problem in L2 and H -~, which will be derived in Sect. 4. In Sect. 3 we discuss the cubic elements and the correspond- ing projections. In Sect. 5 a hybridization of our mixed method is introduced, and its properties with respect to linear algebra and seperconvergence for the scalar variable are studied. In Sect. 6 an Arrow-Hurwitz-type alternating-direc- tion iterative technique is described briefly.