, 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
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
, 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
, 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
,
, 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
Error estimates for Euler forward scheme related to two-phase Stefan problems, RAIRO Modél. Math. Anal. Numér, vol.26, issue.2, pp.365-383, 1992. ,
Some results in lumped mass finite-element approximation of eigenvalue problems using numerical quadrature formulas, J. Comput. Appl. Math, vol.43, issue.3, pp.291-311, 1992. ,
Well-posedness results for triply nonlinear degenerate parabolic equations, J. Differential Equations, vol.247, issue.1, pp.277-302, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-00475758
Global superconvergence and a posteriori error estimates of the finite element method for second-order quasilinear elliptic problems, J. Comput. Appl. Math, vol.260, pp.78-90, 2014. ,
Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal, vol.147, issue.4, pp.269-361, 1999. ,
Expanded mixed finite element methods for quasilinear second order elliptic problems, II. RAIRO Modél. Math. Anal. Numér, vol.32, issue.4, pp.501-520, 1998. ,
The finite element method for elliptic problems, Studies in Mathematics and its Applications, vol.4, 1978. ,
Higher order triangular finite elements with mass lumping for the wave equation, SIAM J. Numer. Anal, vol.38, issue.6, pp.2047-2078, 2001. ,
URL : https://hal.archives-ouvertes.fr/hal-01010373
Nonlinear functional analysis, 1985. ,
A hybrid high-order method for leray-lions elliptic equations on general meshes, Math. Comp, vol.86, issue.307, pp.2159-2191, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01183484
Finite volume schemes for diffusion equations: introduction to and review of modern methods, Special issue on Recent Techniques for PDE Discretizations on Polyhedral Meshes, vol.24, pp.1575-1619, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00813613
The gradient discretisation method, Mathematics & Applications, vol.82, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01382358
Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal, vol.7, issue.1, pp.61-71, 1987. ,
TP or not TP, that is the question, Comput. Geosci, vol.18, issue.3-4, pp.285-296, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00801648
Error estimate for approximate solutions of a nonlinear convectiondiffusion problem, Adv. Differential Equations, vol.7, issue.4, pp.419-440, 2002. ,
Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math, vol.92, issue.1, pp.41-82, 2002. ,
New higher-order mass-lumped tetrahedral elements for wave propagation modelling, SIAM J. Sci. Comput, vol.40, issue.5, pp.2830-2857, 2018. ,
Benchmark on discretization schemes for anisotropic diffusion problems on general grids, Finite volumes for complex applications V, pp.659-692, 2008. ,
URL : https://hal.archives-ouvertes.fr/hal-00429843
Arbitrary high-order finite element schemes and high-order mass lumping, Int. J. Appl. Math. Comput. Sci, vol.17, issue.3, pp.375-393, 2007. ,
URL : https://hal.archives-ouvertes.fr/hal-00603310
Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp, vol.44, issue.170, pp.303-320, 1985. ,
Mass lumping for the optimal control of elliptic partial differential equations, SIAM J. Numer. Anal, vol.55, issue.3, pp.1412-1436, 2017. ,
Leprobì eme de Dirichlet pour leséquationsleséquations elliptiques du second ordrè a coefficients discontinus ,
, Ann. Inst. Fourier, vol.15, pp.189-258, 1965.
Error estimates of the lumped mass finite element method for semilinear elliptic problems, J. Comput. Appl. Math, vol.236, issue.7, pp.1993-2004, 2012. ,
E-mail address: jerome.droniou@monash.edu (R. Eymard) Laboratoire d'Analyse et de Mathématiques Appliquées ,