Interpolation error estimates in W1, p for degenerate Q1 isoparametric elements

For convex quadrilateral elements and 1 ≤ p, the usual W 1,p error estimate for the Q1 isoparametric Lagrange interpolation, called hereafter Q, reads ‖u−Qu‖Lp(K) + h|u−Qu|1,p,K ≤ Ch|u|2,p,K (1) where h denotes the diameter of K. Two facts, about (1), are well known: the convexity of K is a sufficient condition to get the estimate ‖u−Qu‖Lp(K) ≤ Ch|u|2,p,K with C bounded independently on the shape of K, however |u−Qu|1,p,K ≤ Ch|u|2,p,K (2) requires extra assumptions on K to keep C uniformly bounded. In the early work by Ciarlet and Raviart [2] is proved that (2) holds for regular elements if the interior angles of K are bounded from 0 and π. Since then, a large number of geometrical conditions have appeared in the literature and (2) has been proved for different kinds of ”degenerate” elements. As far as we know, for the case p = 2, the most general condition under which (2) holds is that defined by the so called ”regular decomposition property” (or shortly RDP) (see [1]). In this talk we review most of the results concerning (2) and we show that the same conclusions obtained in [1] are valid in W 1,p for 1 ≤ p < 3. We also give counterexamples for the case 3 ≤ p showing that the result can not be generalized for more regular functions. Despite this fact, we prove that optimal order error estimates of the type (2) are valid for any 1 ≤ p, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio (i.e. without the regularity condition). We also show that this restriction on the interior angles can not be removed for 3 ≤ p. ∗ e-mail: [email protected]