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An average interpolation is introduced for 3-rectangles and tetrahedra, and optimal order error estimates in the H 1 norm are proved. The constant in the estimate depends ''weakly'' (improving the results given in Duran (Math. Comp. 68 (1999) 187-199) on the uniformity of the mesh in each direction. For tetrahedra, the constant also depends on the maximum angle of the element. On the other...
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