Volume 50, Number 3, May-June 2016
Special Issue – Polyhedral discretization for PDE
Page(s) 879 - 904
Published online 23 May 2016
  1. R.A. Adams, Sobolev spaces, Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). [Google Scholar]
  2. B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. [CrossRef] [MathSciNet] [Google Scholar]
  3. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO M2AN 19 (1985) 7–32. [Google Scholar]
  4. A.E. Baran and G. Stoyan, Gauss-Legendre elements: A stable, higher order non-conforming finite element family. Comput. 79 (2007) 1–21. [CrossRef] [Google Scholar]
  5. L. Beirão da Veiga, K. Lipnikov and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. [CrossRef] [Google Scholar]
  6. L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method. Vol. 11 of Modeling, Simulations and Applications, 1st edition. Springer-Verlag, New York (2014). [Google Scholar]
  7. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. [CrossRef] [Google Scholar]
  8. L. Beirão da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. [CrossRef] [Google Scholar]
  9. L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl Sci. 24 (2014) 1541–1573. [Google Scholar]
  10. F.M. Benedetto, S. Berrone, S. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Engrg. 280 (2014) 135–156. [CrossRef] [MathSciNet] [Google Scholar]
  11. S.C. Brenner, Poincaré–Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. [CrossRef] [MathSciNet] [Google Scholar]
  12. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts Appl. Math. Springer-Verlag, New York (1994). [Google Scholar]
  13. F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253 (2013) 455–462. [Google Scholar]
  14. F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. [Google Scholar]
  15. F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  16. F. Brezzi, R.S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. [CrossRef] [EDP Sciences] [Google Scholar]
  17. A. Cangiani, G. Manzini and A. Russo, Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47 (2009) 2612–2637. [CrossRef] [MathSciNet] [Google Scholar]
  18. P.G. Ciarlet, Basic Error Estimates for Elliptic Problems. In Vol. II of Handb. Numer. Anal. North-Holland, Amsterdam (1991) 17–351. [Google Scholar]
  19. M.I. Comodi, The Hellan–Herrmann–Johnson method: some new error estimates and postprocessing. Math. Comput. 52 (1989) 17–29. [CrossRef] [Google Scholar]
  20. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973) 33–75. [Google Scholar]
  21. M. Crouzeix and R.S. Falk, Nonconforming finite elements for the Stokes problem. Math. Comput. 52 (1989) 437–456. [CrossRef] [Google Scholar]
  22. D.A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Engrg. 283 (2015) 1–21. [Google Scholar]
  23. D.A. Di Pietro and S. Lemaire, An extension of the Crouzeix-Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84 (2015) 1–31. [CrossRef] [Google Scholar]
  24. D.A. Di Pietro, A. Ern and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14 (2014) 461–472. [CrossRef] [MathSciNet] [Google Scholar]
  25. R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529–550. [CrossRef] [MathSciNet] [Google Scholar]
  26. M. Fortin, A three-dimensional quadratic nonconforming element. Numer. Math. 46 (1985) 269–279. [CrossRef] [MathSciNet] [Google Scholar]
  27. M. Fortin and M. Soulie, A non-conforming piecewise quadratic finite element on triangles. Int. J. Numer. Methods Eng. 19 (1983) 505–520. [Google Scholar]
  28. T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79 (2010) 2169–2189. [CrossRef] [MathSciNet] [Google Scholar]
  29. J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275–311. [Google Scholar]
  30. K. Lipnikov and G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. Accepted for publication in J. Comput. Phys. [Google Scholar]
  31. K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa, The mimetic finite difference method for 3D magnetostatics fields problems. J. Comput. Phys. 230 (2011) 305–328. [CrossRef] [MathSciNet] [Google Scholar]
  32. K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. 257 (2014) 1163–1227. [Google Scholar]
  33. L.D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart–Thomas mixed method. SIAM J. Numer. Anal. 22 (1985) 493–496. [CrossRef] [MathSciNet] [Google Scholar]
  34. G. Matthies and L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. [CrossRef] [MathSciNet] [Google Scholar]
  35. D. Mora, G. Rivera and R. Rodriguez, A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25 (2015) 1421–1445. [CrossRef] [Google Scholar]
  36. C. Ortner, Nonconforming finite-element discretization of convex variational problems. IMA J. Numer. Anal. 31 (2011) 847–864. [CrossRef] [MathSciNet] [Google Scholar]
  37. A. Palha, P.P. Rebelo, R. Hiemstra, J. Kreeft and M. Gerritsma, Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys. 257 (2014) 1394–1422. [Google Scholar]
  38. R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differ. Equ. 8 (1992) 97–111. [Google Scholar]
  39. G. Stoyan and A.E. Baran, Crouzeix-Velte decompositions for higher-order finite elements. Comput. Math. Appl. 51 (2006) 967–986. [CrossRef] [MathSciNet] [Google Scholar]
  40. G. Strang, Variational Crimes in the Finite Element Method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Proc. of Sympos., Univ. Maryland, Baltimore, Md. Academic Press, New York (1972) 689–710. [Google Scholar]
  41. G. Strang and G.J. Fix, An analysis of the finite element method. Prentice-Hall Series in Automatic Computation. Prentice-Hall Inc., Englewood Cliffs, N. J. (1973). [Google Scholar]
  42. R. Verfürth. A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715–719. [CrossRef] [EDP Sciences] [Google Scholar]

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