Summary. Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates in L2(t?) and H-~(O) are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.

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Abstract. A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as tends to zero.

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A mixed finite element procedure for plane elasticity is int. ,duced and analyzed. The symmetry of the stress tensor is enforced through the introduction of a Lagrange multiplier. An additional Lagrange multiplier is introduced to simplify the linear algebraic system. Applications are made to incompressible elastic problems and to plasticity problems.

Abstract. Global error estimates in L2(Q), L°°(Q), and H~S(Q), Q in R2 or R3, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator Lp = -div(a grad p + bp) + cp based on the Raviart-Thomas-Nedelec space V^ X Wh c H(div; Í2) X L2(ü). Optimal order estimates are obtained for the approximation of p and the associated velocity field u = -(a grad p + bp) in L2(fl) and H~S(Q), 0 < s < k + 1, and, if ß c R2, for/? in L°°(Q)..

Summary. Two families of mixed finite elements, one based on simplices and the other on cubes, are introduced as alternatives to the usual RaviartThomas-Nedelec spaces. These spaces are analogues of those introduced by Brezzi, Douglas, and Marini in two space variables. Error estimates in L 2 and H -s are derived.