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.

If 12 C R" is a bounded domain, the existence of solutions u e U'/ r (12) of divu = f for f e ¿-^(12) with vanishing mean value and 1 < p < oo. is a basic result in the analysis of the Stokes equations. It is known that the result holds when 12 is a Lipschitz domain and that it is not valid for domains with external cusps. In this paper we prove that the result holds for John domains. Our proof is constructive: the so lution u is given by an explicit integral operator acting on f. To prove that u e VK0'A12) we make use of the Calderon-Zygmund singular integral operator theory and the Hardy-Littlewood maximal function. For domains satisfying the separation property introduced in [S. Buckley. P. Koskela, Sobolev-Poincare implies John. Math. Res. Lett. 2 (5) (1995) 577-593], and 1 < p < n. we also prove a converse result, thus characterizing in this case the domains for which a continuous right inverse of

For the Lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition. The objective of this paper is to obtain similar results for the Raviart-Thomas interpolation arising in the analysis of mixed methods. We prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and, moreover, that this condition is indeed necessary to have these estimates. Error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the Stokes equations are derived from our results. AMS subject classifications. 65N15, 65N30PII. S0036142997331293 1. Introduction. This paper deals with the convergence of nonconforming and mixed finite elements when narrow (or degenerate) elements are used in two and three dimensions. The use of these kinds of elements is of interest in many situations, in particular for nonisotropic problems involving solutions which behave differently in each direction. We obtain optimal order error estimates for the Raviart-Thomas (RT) interpolation of the lowest degree for triangular and tetrahedral elements satisfying a maximum angle condition. Our arguments also apply to degenerate rectangular elements providing error estimates with constants independent of the relation between the mesh size in each direction.We give two applications of our result. The first one follows immediately from the known error analysis of RT mixed approximation of second order elliptic problems. Indeed, it is known [20, 10] that the error in that approximation can be bounded in terms of the error for the RT interpolation. Therefore, our results give optimal error estimates for degenerate elements.Our second application concerns the approximation of the Stokes equations by the linear nonconforming elements of Crouzeix and Raviart [9].The standard finite element analysis requires the so-called regularity assumption on the elements, i.e., bounded ratio between outer and inner diameter (see, for example, [7,6]). This same hypothesis is needed for the analysis of mixed methods based on the Piola transform given in [20] and for the analysis of nonconforming approximations given in [9].However, for standard finite element approximations of coercive problems (such as scalar second order elliptic problems) it is now well known that the regularity assumption can be relaxed. Indeed, for two-dimensional (2D) problems, optimal order

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