E. Eymard, the theorem. Finally, the H 2 regularity property is a straightforward consequence of the optimal elliptic regularity on convex domains for Lipschitz-continuous diffusion tensor

A. Lemma, Under Assumption (1.3a 1.3a), there exists a sequence (D m ) m?N = (X D m ,0 , ? D m , D m , Q D m ) m?N of gradient discretisations

, such that max T ? Mm diam(T ) ? 0 and (M m ) m?N is regular in the sense that the ratio of the diameter of T ? M m over the largest ball inside T is bounded uniformly with respect to T and m. We let M m = {T ? M m : T ? ?} and define the polyhedral set ? m ? ? as the interior of ? T ?Mm T m. The gradient discretiation D m = (X D m ,0 , ? D m , D m , Q D m ) is defined as the mass-lumped conforming P 1 gradient discretisation on the mesh M m of ? m, vol.12, p.12

?. For-v-?-x-d-m, D m v) |?i = v i for all i ? V m , where (? i ) i?Vm is the dual (Donald) mesh of M m , and ? D m v = 0 on ?\? m, p.0

?. For-v-?-x-d-m, D m v is on ? m the gradient of the conforming P 1 reconstruction from the vertex values (v i ) i?Vm , and D m v = 0 on ?\? m

?. Q-d-m-=-id,

, The properties of mass-lumped P 1 GDs on ? m (see [12 12, Theorem 8.17]) then show that (D m ) m?N is coercive, limit-conforming and compact, Since the functions and gradient reconstructions are extended by 0 outside ? m , C D m and W D m can be computed using norms and integrals over ? m

, that case, for m large enough, ? ? C ? c (? m ) and the norms in S D m (?) can be restricted to ? m

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(. J. Droniou, E-mail address: jerome.droniou@monash.edu (R. Eymard) Laboratoire d'Analyse et de Mathématiques Appliquées