Issue
ESAIM: M2AN
Volume 50, Number 3, May-June 2016
Special Issue – Polyhedral discretization for PDE
Page(s) 727 - 747
DOI https://doi.org/10.1051/m2an/2015067
Published online 23 May 2016
  1. 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. [Google Scholar]
  2. P.F. Antonietti, L. Beirão da Veiga, D. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52 (2014) 386–404. [Google Scholar]
  3. P.F. Antonietti, N. Bigoni and M. Verani, Mimetic discretizations of elliptic control problems. J. Sci. Comput. 56 (2013) 14–27. [Google Scholar]
  4. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749–1779. [Google Scholar]
  5. M. Arroyo and M. Ortiz, Local maximum-entropy approximation schemes, Meshfree methods for partial differential equations III. Vol. 57 of Lect. Notes Comput. Sci. Engrg. Springer, Berlin (2007) 1–16. [Google Scholar]
  6. I. Babuška and J.E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. [CrossRef] [MathSciNet] [Google Scholar]
  7. I. Babuška and J.M. Melenk, The partition of unity method. Int. J. Numer. Methods Engrg. 40 (1997) 727–758. [CrossRef] [MathSciNet] [Google Scholar]
  8. I. Babuška, U. Banerjee and J.E. Osborn, Generalized finite element methods – main ideas, results and perspective. Int. J. Comput. Methods 01 (2004) 67–103. [CrossRef] [Google Scholar]
  9. L. Beirão da Veiga , A residual based error estimator for the mimetic finite difference method. Numer. Math. 108 (2008) 387–406. [Google Scholar]
  10. L. Beirão da Veiga, and G. Manzini, A higher-order formulation of the mimetic finite difference method. SIAM J. Sci. Comput. 31 (2008) 732–760. [CrossRef] [MathSciNet] [Google Scholar]
  11. L. Beirão da Veiga and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781. [Google Scholar]
  12. L. Beirão da Veiga, K. Lipnikov and G. Manzini, Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325–356. [CrossRef] [MathSciNet] [Google Scholar]
  13. L. Beirão da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys. 228 (2009) 7215–7232. [Google Scholar]
  14. 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. [Google Scholar]
  15. 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. [Google Scholar]
  16. 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]
  17. 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]
  18. L. Beirão da Veiga, K. Lipnikov and G. Manzini, The mimetic finite difference method for elliptic problems. Vol. 11 of MS&A. Model. Simul. Appl.. Springer–Verlag (2014). [Google Scholar]
  19. L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, H(div) and H(curl)-conforming virtual element methods. Numer. Math. Doi:10.1007/s00211-015-0746-1 (2015). [Google Scholar]
  20. L Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, Virtual Element Methods for general second order-elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016) 729. [Google Scholar]
  21. M.F. 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. [Google Scholar]
  22. S.O.R. Biabanaki, A.R. Khoei and P. Wriggers, Polygonal finite element methods for contact-impact problems on non-conformal meshes. Comp. Methods Appl. Mech. Engrg. 269 (2014) 198–221. [Google Scholar]
  23. J.E. Bishop, A displacement-based finite element formulation for general polyhedra using harmonic shape functions. Int. J. Numer. Methods Engrg. 97 (2014) 1–31. [CrossRef] [Google Scholar]
  24. P.B. Bochev and J.M. Hyman, Principles of Mimetic Discretizations of Differential Operators. Compatible Spatial Discretizations. Vol. 142 of IMA Volumes Math. Appl. Springer, New York (2006) 89–119. [Google Scholar]
  25. J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN 48 (2014) 553–581. [CrossRef] [EDP Sciences] [Google Scholar]
  26. F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253 (2013) 455–462. [Google Scholar]
  27. F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 1533–1551. [Google Scholar]
  28. F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16 (2006) 275–297. [CrossRef] [MathSciNet] [Google Scholar]
  29. F. Brezzi, K. Lipnikov, M. Shashkov and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3682–3692. [Google Scholar]
  30. 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]
  31. F. Brezzi, A. Buffa, K. Lipnikov and G. Manzini, The mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes. J. Comput. Phys. 230 (2011) 305–328. [CrossRef] [MathSciNet] [Google Scholar]
  32. 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]
  33. A. Cangiani, G. Manzini, A. Russo and N. Sukumar, Hourglass stabilization and the virtual element method. Int. J. Numer. Methods Engrg. 102 (2015) 404–436. [Google Scholar]
  34. J. Chessa, P. Smolinski and T. Belytschko, The extended finite element method (XFEM) for solidification problems. Int. J. Numer. Methods Engrg. 53 (2002) 1959–1977. [CrossRef] [Google Scholar]
  35. H. Chi, C. Talischi, O. Lopez-Pamies and G.H. Paulino, Polygonal finite elements for finite elasticity. Int. J. Numer. Methods Engrg. 101 (2015) 305–328. [CrossRef] [Google Scholar]
  36. P.G. Ciarlet, The finite element method for elliptic problems. Vol. 4 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978). [Google Scholar]
  37. B. Cockburn, The hybridizable discontinuous Galerkin methods. In Vol. IV of Proc. of the International Congress of Mathematicians. Hindustan Book Agency, New Delhi (2010), 2749–2775. [Google Scholar]
  38. B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. [Google Scholar]
  39. B. Cockburn, J. Guzmán and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comp. 78 (2009) 1–24. [Google Scholar]
  40. B. Cockburn, J. Gopalakrishnan and F.-J. Sayas, A projection-based error analysis of HDG methods. Math. Comp. 79 (2010) 1351–1367. [Google Scholar]
  41. D. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. Vol. 69 of Math. Appl. Springer, Heidelberg (2012). [Google Scholar]
  42. D. Di Pietro and A. Ern, A family of arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. Available at https://hal.archives-ouvertes.fr/hal-00918482 (2013). [Google Scholar]
  43. D. 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. [Google Scholar]
  44. D. Di Pietro and A. Alexandre 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]
  45. D. Di Pietro and A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Acad. Sci. Paris, Ser. I 353 (2015), 31–34. [Google Scholar]
  46. J. Douglas Jr., and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comp. 44 (1985) 39–52. [CrossRef] [MathSciNet] [Google Scholar]
  47. Jerome Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. [CrossRef] [MathSciNet] [Google Scholar]
  48. J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010) 265–295. [Google Scholar]
  49. J. Droniou, R. Eymard, T. Gallouët and R. Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. 23 (2013) 2395–2432. [Google Scholar]
  50. Q. Du, V. Faber and M. Gunzburger, Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41 (1999) 637–676. [Google Scholar]
  51. M. Floater, A. Gillette and N. Sukumar, Gradient bounds for Wachspress coordinates on polytopes. SIAM J. Numer. Anal. 52 (2014) 515–532. [CrossRef] [MathSciNet] [Google Scholar]
  52. M.S. Floater, G. Kós and M. Reimers, Mean value coordinates in 3d. Comput. Aided Geom. Design 22 (2005) 623–631. [CrossRef] [MathSciNet] [Google Scholar]
  53. M. Floater, K. Hormann and G. Kós, A general construction of barycentric coordinates over convex polygons. Adv. Comput. Math. 24 (2006) 311–331. [CrossRef] [Google Scholar]
  54. T.-P. Fries and T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Engrg. 84 (2010) 253–304. [Google Scholar]
  55. A.L. Gain, Polytope-based topology optimization using a mimetic-inspired method. Ph.D. thesis, University of Illinois at Urbana-Champaign, 2013. [Google Scholar]
  56. A.L. Gain, C. Talischi and G.H. Paulino, On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 282 (2014) 132–160. [Google Scholar]
  57. A. Gerstenberger and W.A. Wall, An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1699–1714. [CrossRef] [MathSciNet] [Google Scholar]
  58. K. Hormann and M.S. Floater, Mean value coordinates for arbitrary planar polygons. ACM Trans. Graph. 25 (2006) 1424–1441. [CrossRef] [Google Scholar]
  59. S.R. Idelsohn, E. Oñate, N. Calvo and F. Del Pin, The meshless finite element method. Int. J. Numer. Methods Engrg. 58 (2003) 893–912. [CrossRef] [Google Scholar]
  60. K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. 257 (2014) 1163–1227. [Google Scholar]
  61. G. Manzini, A. Russo and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods. Math. Models Methods Appl. Sci. 24 (2014) 1665–1699. [CrossRef] [MathSciNet] [Google Scholar]
  62. S. Martin, P. Kaufmann, M. Botsch, M. Wicke and M. Gross, Polyhedral finite elements using harmonic basis functions. Comput. Graph. Forum 27 (2008) 1521–1529. [CrossRef] [Google Scholar]
  63. J.M. Melenk and I. Babuska, The partition of unity finite element method: basic theory and applications. Comp. Methods Appl. Mech. Engrg. 139 (1996) 289–314. [Google Scholar]
  64. R. Merle and J. Dolbow, Solving thermal and phase change problems with the extended finite element method. Comput. Mech. 28 (2002) 339–350. [CrossRef] [Google Scholar]
  65. S. Mohammadi, Extended Finite Element Method. Blackwell Publishing Ltd (2008). [Google Scholar]
  66. D. Mora, G. Rivera and R. Rodríguez, A virtual element method for the Steklov eigenvalue problem. Math. Models Meth. Appl. Math. 25 (2015) 1421–1445. [Google Scholar]
  67. L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao, Weak Galerkin methods for second order elliptic interface problems. J. Comput. Phys. 250 (2013) 106–125. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  68. L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction. J. Comput. Appl. Math. 285 (2015) 45–58. [CrossRef] [MathSciNet] [Google Scholar]
  69. N. C. Nguyen, J. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous galerkin method for linear convection-diffusion equations. J. Comput. Phys. 228 (2009) 3232–3254. [CrossRef] [MathSciNet] [Google Scholar]
  70. J. Oswald, R. Gracie, R. Khare and T. Belytschko, An extended finite element method for dislocations in complex geometries: Thin films and nanotubes. Comp. Methods Appl. Mech. Engrg. 198 (2009) 1872–1886. [CrossRef] [Google Scholar]
  71. T. Rabczuk, S. Bordas and G. Zi, On three-dimensional modelling of crack growth using partition of unity methods. Special Issue: Association of Computational Mechanics United Kingdom. Comput. Struct. 88 (2010) 1391–1411. [CrossRef] [Google Scholar]
  72. A. Rand, A. Gillette and C. Bajaj, Interpolation error estimates for mean value coordinates over convex polygons. Adv. Comput. Math. 39 (2013) 327–347. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  73. S. Rjasanow and S. Weisser, FEM with Trefftz trial functions on polyhedral elements. J. Comp. Appl. Math. 263 (2014) 202–217. [Google Scholar]
  74. B.G. Smith, B.L. Jr. Vaughan and D.L. Chopp, The extended finite element method for boundary layer problems in biofilm growth. Commun. App. Math. Comp. Sci. 2 (2007) 35–56. [CrossRef] [Google Scholar]
  75. M. Spiegel et al., Tetrahedral vs. polyhedral mesh size evaluation on flow velocity and wall shear stress for cerebral hemodynamic simulation. Comput. Meth. Biomech. Biomed. Engrg. 14 (2011) 9–22. [Google Scholar]
  76. N. Sukumar, Construction of polygonal interpolants: a maximum entropy approach. Int. J. Numer. Methods Engrg. 61 (2004) 2159–2181. [CrossRef] [Google Scholar]
  77. N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Methods Engrg. 61 (2004) 2045–2066. [Google Scholar]
  78. N. Sukumar and E.A. Malsch, Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Engrg. 13 (2006) 129–163. [CrossRef] [MathSciNet] [Google Scholar]
  79. N. Sukumar, N. Möes, B. Moran and T. Belytschko, Extended finite element method for three-dimensional crack modelling. Int. J. Numer. Methods Eng. 48 (2000) 1549–1570. [CrossRef] [Google Scholar]
  80. N. Sukumar, D.L. Chopp, N. Möes and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite-element method. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6183–6200. [CrossRef] [MathSciNet] [Google Scholar]
  81. A. Sutradhar, G.H. Paulino, M.J. Miller and T.H. Nguyen, Topology optimization for designing patient-specific large craniofacial segmental bone replacements. Proc. Natl. Acad. Sci. USA 107 (2010) 13222–13227. [CrossRef] [Google Scholar]
  82. C. Talischi and G.H. Paulino, Addressing integration error for polygonal finite elements through polynomial projections: a patch test connection. Math. Models Methods Appl. Sci. 24 (2014) 1701–1727. [CrossRef] [Google Scholar]
  83. C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes, Polygonal finite elements for topology optimization: A unifying paradigm. Int. J. Numer. Methods Engrg. 82 (2010) 671–698. [Google Scholar]
  84. L.M. Vigneron, J.G. Verly and S.K. Warfield, On extended finite element method (XFEM) for modelling of organ deformations associated with surgical cuts. Edited by S. Cotin and D. Metaxas, Medical Simulation, Vol. 3078 of Lect. Notes Comput. Sci. Springer, Berlin (2004). [Google Scholar]
  85. E. Wachspress, Ed., A Rational Finite Element Basis. Vol. 114 of Math. Sci. Engrg. Academic Press, Inc., New York, London (1975). [Google Scholar]
  86. G.J. Wagner, N. Möes, W.K. Liu and T. Belytschko, The extended finite element method for rigid particles in stokes flow. Int. J. Numer. Methods Engrg. 51 (2001) 293–313. [Google Scholar]
  87. J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241 (2013) 103–115. [CrossRef] [MathSciNet] [Google Scholar]
  88. J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comp. 83 (2014) 2101–2126. [CrossRef] [MathSciNet] [Google Scholar]
  89. J. Warren, Barycentric coordinates for convex polytopes. Adv. Comput. Math. 6 (1996) 97–108. [CrossRef] [MathSciNet] [Google Scholar]

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