Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 20, 2016

Convergence analysis of a finite element method for second order non-variational elliptic problems

Michael Neilan


We introduce and analyze a family of finite element methods for elliptic partial differential equations in non-variational form with non-differentiable coefficients. The finite element method studied is a variant of the one recently proposed in [Lakkis & Pryer, SIAM J. Sci. Comput., 2011], where a finite element Hessian is introduced as an auxiliary unknown. We modify the definition of the finite element Hessian rendering the auxiliary variable completely local, thus resulting in a more efficient scheme. We show that the method is stable under general conditions on the coefficient matrix and derive error estimates in a discrete H2-norm provided the discretization parameter is sufficiently small. Numerical experiments are presented which verify the theoretical results.

JEL Classification: 65N30; 65N12; 35J25

Funding statement: Supported in part by NSF grant DMS–1417980 and the Alfred Sloan Foundation


[1] R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York–London, 1975.Search in Google Scholar

[2] P.R. Amestoy, I.S. Duff, and J.-Y. L’Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods in Appl. Mech. Eng., 184 (2000), 501–520.10.1016/S0045-7825(99)00242-XSearch in Google Scholar

[3] I. Babuska, G. Caloz, and J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31(4) (1994), 945–981.10.1137/0731051Search in Google Scholar

[4] S. Bernstein, Sur la généralisation du probléme de Dirichlet, Math. Ann., 69(1) (1910), 82–136.10.1007/BF01455154Search in Google Scholar

[5] S. C. Brenner, and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition, Springer, 2008.10.1007/978-0-387-75934-0Search in Google Scholar

[6] S.C. Brenner and L.-Y. Sung, C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput., 22/23 (2005), 83–118.10.1007/s10915-004-4135-7Search in Google Scholar

[7] S.C. Brenner, T. Gudi, M. Neilan, and L.-Y. Sung, C0 penalty methods for the fully nonlinear Monge–Ampère equation, Math. Comp., 80 (2011), 1979–1995.10.1090/S0025-5718-2011-02487-7Search in Google Scholar

[8] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.10.1115/1.3424474Search in Google Scholar

[9] X. Feng, L. Hennings, and M. Neilan, C0discontinuous Galerkin finite element methods for second order linear elliptic partial differential equations in non-divergence form, arXiv:1505.02842 [math.NA].Search in Google Scholar

[10] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer, 2006.Search in Google Scholar

[11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.10.1007/978-3-642-61798-0Search in Google Scholar

[12] J. Huang, X. Huang, and W. Han, A new C0 discontinuous Galerkin method for Kirchhoff plates, Comput. Methods Appl. Mech. Engrg., 199(23–24) (2010), 1446–1454.10.1016/j.cma.2009.12.012Search in Google Scholar

[13] M. Jensen and I. Smears, On the convergence of finite element methods for Hamilton–Jacobi–Bellman equations, SIAM J. Numer. Anal., 51(1) (2013), 137–162.10.1137/110856198Search in Google Scholar

[14] O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon Ehrenpreis. Academic Press, New York–London, 1968.Search in Google Scholar

[15] O. Lakkis and T. Pryer, A finite element method for second order nonvariational elliptic problems, SIAM J. Sci. Comput., 33(2) (2011), 786–801.10.1137/100787672Search in Google Scholar

[16] M. Neilan, Quadratic finite element methods for the Monge–Ampère equation, J. Sci. Comput., 54(1) (2013), 200–226.10.1007/s10915-012-9617-4Search in Google Scholar

[17] J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz–Galerkin methods, Math. Comp., 28 (1974), 937–958.10.1090/S0025-5718-1974-0373325-9Search in Google Scholar

[18] R.H. Nochetto and Wujun Zhang, Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form, arXiv:1411.6036 [math.NA], 2014.Search in Google Scholar

[19] A. H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates, Math. Comp., 67(223) (1998), 877–899.10.1090/S0025-5718-98-00959-4Search in Google Scholar

[20] A.H. Schatz and J. Wang, Some new error estimates for Ritz–Galerkin methods with minimal regularity assumptions, Math. Comp., 65(213) (1996), 19–27.10.1090/S0025-5718-96-00649-7Search in Google Scholar

[21] S. Zhang, A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math., 59(1) (2009), 219–233.10.1016/j.apnum.2008.02.002Search in Google Scholar

[22] I. Smears and E. Süli, Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordès coefficients, SIAM J. Numer. Anal., 51(4) (2013), 2088–2106.10.1137/120899613Search in Google Scholar

Received: 2016-3-10
Revised: 2016-3-28
Accepted: 2016-4-9
Published Online: 2016-5-20
Published in Print: 2017-9-26

© 2016 by Walter de Gruyter Berlin/Boston

Scroll Up Arrow