In this paper we prove error estimates for a piecewise Q 1 average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions. Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses. Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems. Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements. As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.

Abstract. We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra.Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree k ≥ 0, we prove error estimates of order j + 1 when the vector field being approximated has components in W j+1,p , for triangles or tetrahedra, where 0 ≤ j ≤ k and 1 ≤ p ≤ ∞. These results are new even in the two-dimensional case. Indeed, the estimate was known only in the case j = k. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for k = 0.

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