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All issues Volume 50 / No 3 (May-June 2016) ESAIM: M2AN, 50 3 (2016) 651-676 References
Issue
ESAIM: M2AN
Volume 50, Number 3, May-June 2016
Special Issue – Polyhedral discretization for PDE
Page(s) 651 - 676
DOI https://doi.org/10.1051/m2an/2015096
Published online 23 May 2016
  1. G. Acosta, Lagrange and average interpolation over 3D anisotropic elements. J. Comput. Appl. Math. 135 (2001) 91–109. [CrossRef] [MathSciNet] [Google Scholar]
  2. G. Acosta and R.G. Durán, The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37 (2000) 18–36. [Google Scholar]
  3. R.A. Adams, Sobolev Spaces. In vol. 140 of Pure and Applied Mathematics, 2nd edition. Academic Press, Oxford (2003). [Google Scholar]
  4. P. Antonietti, M. Verani and L. Zikatanov, A two-level method for mimetic finite difference discretizations of elliptic problems. Comput. Math. Appl. 70 (2015) 2674–2687. [CrossRef] [MathSciNet] [Google Scholar]
  5. T. Apel, Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. Teubner, Stuttgart (1999). [Google Scholar]
  6. I. Babuška and A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976) 214–226. [CrossRef] [MathSciNet] [Google Scholar]
  7. I. Babuška, U. Banerjee and J. Osborn, Meshless and Generalized Finite Element Methods: A Survey of Some Major Results. In Meshfree Methods for Partial Differential Equations. Springer (2002) 1–20. [Google Scholar]
  8. 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]
  9. 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]
  10. L. Beirão da Veiga, F. Brezzi, L. Marini and A. Russo, The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24 (2014) 1541–1573. [CrossRef] [Google Scholar]
  11. W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts. Math. Annal. 76 (1915) 504–513. [Google Scholar]
  12. P. Bochev and J. Hyman, Principles of Mimetic Discretizations of Differential Operators. In Compatible Spatial Discretizations. Springer (2006) 89–119. [Google Scholar]
  13. 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]
  14. J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for the Stokes equations on polyhedral meshes. IMA J. Numer. Anal. 35 (2015) 1672–1697. [CrossRef] [MathSciNet] [Google Scholar]
  15. J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112–124. [Google Scholar]
  16. J. Brandts, S. Korotov and M. Kzříěk, On the equivalence of ball conditions for simplicial finite elements. Appl. Math. Lett. 22 (2009) 1210–1212. [CrossRef] [MathSciNet] [Google Scholar]
  17. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics. Springer, New York, third edition (2008). [Google Scholar]
  18. F. Brezzi, A. Buffa and G. Manzini, Mimetic scalar products of discrete differential forms. J. Comput. Phys. 257 (2014) 1228–1259. [CrossRef] [MathSciNet] [Google Scholar]
  19. F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Meth. Appl. Sci. 15 (2005) 1533–1551. [Google Scholar]
  20. S. Cai and T.J. Tautges, One-to-one sweeping based on harmonic ST mappings of facet meshes and their cages. Engrg. Comput. (2014) 1–14. [Google Scholar]
  21. A. Cangiani, G. Manzini, A. Russo and N. Sukumar, Hourglass stabilization and the virtual element method. Int. J. Numer. Meth. Engng 1 (2014) 1–33. [Google Scholar]
  22. L. Chen and J. Xu, Optimal Delaunay triangulation. J. Comput. Math. 22 (2004) 299–308. [MathSciNet] [Google Scholar]
  23. S.-W. Cheng, T.K. Dey, H. Edelsbrunner, M.A. Facello and S.-H. Teng, Sliver exudation. J. ACM 47 (2000) 883–904. [CrossRef] [MathSciNet] [Google Scholar]
  24. L.P. Chew, Constrained Delaunay triangulations. Algorithmica 4 (1989) 97–108. [CrossRef] [MathSciNet] [Google Scholar]
  25. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 40 of Classics in Applied Mathematics, 2nd edition. SIAM, Philadelphia, PA (2002). [Google Scholar]
  26. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19 (1988) 613–626. [CrossRef] [MathSciNet] [Google Scholar]
  27. E.F. D’Azevedo and R.B. Simpson, On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10 (1989) 1063–1075. [CrossRef] [Google Scholar]
  28. S. Dekel and D. Leviatan, The Bramble–Hilbert lemma for convex domains. SIAM J. Math. Anal. 35 (2004) 1203–1212. [CrossRef] [MathSciNet] [Google Scholar]
  29. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. [CrossRef] [MathSciNet] [Google Scholar]
  30. Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Amer. Math. Soc. 124 (1996) 591–600. [Google Scholar]
  31. R.G. Durán, Error estimates for 3-d narrow finite elements. Math. Comput. 68 (1999) 187–199. [CrossRef] [Google Scholar]
  32. H. Edelsbrunner and D. Guoy, An experimental study of sliver exudation. Engrg. With Comput. 18 (2002) 229–240. [CrossRef] [Google Scholar]
  33. H. Edelsbrunner, X.-Y. Li, G. Miller, A. Stathopoulos, D. Talmor, S.-H. Teng, A. Üngör and N. Walkington. Smoothing and cleaning up slivers. In Proc. of ACM Symp. Theory Comput. (2000) 273–277. [Google Scholar]
  34. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). [Google Scholar]
  35. T. Euler, R. Schuhmann and T. Weiland, Polygonal finite elements. IEEE Trans. Magnetics 42 (2006) 675–678. [CrossRef] [Google Scholar]
  36. L.C. Evans, Partial Differential Equations. Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998). [Google Scholar]
  37. L.C. Evans and R.F. Gariepy. Measure theory and fine properties of functions. In vol. 5. CRC press (1991). [Google Scholar]
  38. M. Floater, Mean value coordinates. Comput. Aided Geom. Des. 20 (2003) 19–27. [CrossRef] [Google Scholar]
  39. 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]
  40. M. Floater, G. Kós and M. Reimers, Mean value coordinates in 3D. Comput. Aided Geom. Des. 22 (2005) 623–631. [Google Scholar]
  41. A. Gillette, A. Rand and C. Bajaj, Error estimates for generalized barycentric coordinates. Adv. Comput. Math. 37 (2012) 417–439. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  42. S. Guattery, G.L. Miller and N. Walkington, Estimating interpolation error: A combinatorial approach. In Proc. of 10th Symp. Discrete Algorithms (1999) 406–413. [Google Scholar]
  43. A. Hannukainen, S. Korotov and M. Kzříěk, The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120 (2012) 79–88. [CrossRef] [MathSciNet] [Google Scholar]
  44. K. Hormann and N. Sukumar, Maximum entropy coordinates for arbitrary polytopes. Comput. Graphics Forum 27 (2008) 1513–1520. [CrossRef] [Google Scholar]
  45. P. Jamet, Estimations d’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10 (1976) 43–61. [Google Scholar]
  46. P. Joshi, M. Meyer, T. DeRose, B. Green and T. Sanocki, Harmonic coordinates for character articulation. ACM Trans. Graphics 26 (2007) 71. [Google Scholar]
  47. T. Ju, P. Liepa and J. Warren, A general geometric construction of coordinates in a convex simplicial polytope. Comput. Aided Geom. Des. 24 (2007) 161–178. [CrossRef] [Google Scholar]
  48. T. Ju, S. Schaefer and J. Warren, Mean value coordinates for closed triangular meshes. ACM Trans. Graphics 24 (2005) 561–566. [CrossRef] [Google Scholar]
  49. K. Kobayashi and T. Tsuchiya, A Babuška-Aziz type proof of the circumradius condition. Jpn J. Ind. Appl. Math. 31 (2014) 193–210. [CrossRef] [MathSciNet] [Google Scholar]
  50. K. Kobayashi and T. Tsuchiya, On the circumradius condition for piecewise linear triangular elements. Japan J. Ind. Appl. Math. 32 (2015) 65–76. [CrossRef] [Google Scholar]
  51. M. Kzříěk, On semiregular families of triangulations and linear interpolation. Appl. Math. 36 (1991) 223–232. [MathSciNet] [Google Scholar]
  52. M. Kzříěk, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992) 513–520. [CrossRef] [MathSciNet] [Google Scholar]
  53. T. Lambert, The Delaunay Triangulation Maximizes the Mean Inradius. In Proc. of Canadian Conf. Comput. Geom. Citeseer (1994) 201–206. [Google Scholar]
  54. C.L. Lawson, Software for C1 Interpolation. In Mathematical Software III, edited by J.R. Rice (1977) 161–194. [Google Scholar]
  55. A. Lemoine, J.-P. Caltagirone, M. Azaïez and S. Vincent, Discrete Helmholtz–Hodge decomposition on polyhedral meshes using compatible discrete operators. J. Sci. Comput. 65 (2015) 34–53. [CrossRef] [MathSciNet] [Google Scholar]
  56. N.J. Lennes, Theorems on the simple finite polygon and polyhedron. Amer. J. Math. (1911) 37–62. [Google Scholar]
  57. G. Leoni, A First Course in Sobolev Spaces. Vol. 105 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2009). [Google Scholar]
  58. S. Li and W.K. Liu, Meshfree and particle methods and their applications. Appl. Mech. Rev. 55 (2002) 1–34. [CrossRef] [Google Scholar]
  59. X.-Y. Li, Generating well-shaped d-dimensional Delaunay meshes. Theoret. Comput. Sci. 296 (2003) 145–165. [CrossRef] [MathSciNet] [Google Scholar]
  60. X.-Y. Li and S.-M. Hu, Poisson coordinates. IEEE Trans. Visualization Comput. Graphics 19 (2013) 344–352. [CrossRef] [Google Scholar]
  61. K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. 257 (2014) 1163–1227. [Google Scholar]
  62. 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]
  63. J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains. Manuscripta Math. 58 (1987) 47–65. [CrossRef] [MathSciNet] [Google Scholar]
  64. S. Martin, P. Kaufmann, M. Botsch, M. Wicke and M. Gross, Polyhedral finite elements using harmonic basis functions. In Proc. of Symp. Geom. Proc. (2008) 1521–1529. [Google Scholar]
  65. G.L. Miller, D. Talmor, S.-H. Teng and N. Walkington, On the radius-edge condition in the control volume method. SIAM J. Numer. Anal. (1999) 1690–1708. [Google Scholar]
  66. L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Model. 12 (2015) 31–53. [MathSciNet] [Google Scholar]
  67. P.L. Powar, Minimal roughness property of the Delaunay triangulation: A shorter approach. Comput. Aided Geom. Des. 9 (1992) 491–494. [CrossRef] [Google Scholar]
  68. J. Rambau, On a generalization of Schönhardt’s polyhedron. Combin. Comput. Geom. 52 (2003) 510–516. [Google Scholar]
  69. A. Rand, Delaunay Refinement Algorithms for Numerical Methods. Ph.D. thesis, Carnegie Mellon University (2009). [Google Scholar]
  70. A. Rand, Average interpolation under the maximum angle condition. SIAM J. Numer. Anal. 50 (2012) 2538–2559. [CrossRef] [MathSciNet] [Google Scholar]
  71. A. Rand, A. Gillette and C. Bajaj, Interpolation error estimates for mean value coordinates. Adv. Comput. Math. 39 (2013) 327–347. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  72. S. Rippa, Minimal roughness property of the Delaunay triangulation. Comput. Aided Geom. Des. 7 (1990) 489–497. [CrossRef] [Google Scholar]
  73. S. Rippa, Long and thin triangles can be good for linear interpolation. SIAM J. Numer. Anal. 29 (1992) 257–270. [CrossRef] [MathSciNet] [Google Scholar]
  74. S. Rippa and B. Schiff, Minimum energy triangulations for elliptic problems. Comput. Methods Appl. Mech. Engrg. 84 (1990) 257–274. [CrossRef] [MathSciNet] [Google Scholar]
  75. E. Schönhardt, Über die zerlegung von dreieckspolyedern in tetraeder. Math. Annal. 98 (1928) 309–312. [Google Scholar]
  76. N.A. Shenk, Uniform error estimates for certain narrow Lagrange finite elements. Math. Comput. 63 (1994) 105–119. [CrossRef] [Google Scholar]
  77. J.R. Shewchuk, General-dimensional constrained Delaunay and constrained regular triangulations, I: Combinatorial properties. Discrete Comput. Geom. 39 (2008) 580–637. [CrossRef] [MathSciNet] [Google Scholar]
  78. R. Sibson, Locally equiangular triangulations. Comput. J. 21 (1978) 243–245. [CrossRef] [MathSciNet] [Google Scholar]
  79. R. Sibson, A vector identity for the Dirichlet tessellation. Math. Proc. Cambridge Philos. Soc. 87 (1980) 151–155. [Google Scholar]
  80. N. Sukumar, Construction of polygonal interpolants: a maximum entropy approach. Int. J. Numer. Methods Eng. 61 (2004) 2159–2181. [Google Scholar]
  81. N. Sukumar and R. Wright, Overview and construction of meshfree basis functions: From moving least squares to entropy approximants. Int. J. Numer. Methods Eng. 70 (2007) 181–205. [CrossRef] [Google Scholar]
  82. N. Sukumar, B. Moran, A. Semenov and V. Belikov, Natural neighbour Galerkin methods. Int. J. Numer. Meth. Engng 50 (2001) 1–27. [CrossRef] [Google Scholar]
  83. J.L. Synge, The Hypercircle in Mathematical Physics. Cambridge University Press, Cambridge (1957). [Google Scholar]
  84. A. Tabarraei and N. Sukumar, Application of polygonal finite elements in linear elasticity. Int. J. Comput. Methods 3 (2006) 503–520. [CrossRef] [MathSciNet] [Google Scholar]
  85. R. Verfürth, A note on polynomial approximation in Sobolev spaces. Math. Model. Numer. Anal. 33 (1999) 715–719. [Google Scholar]
  86. E. Wachspress, A Rational Finite Element Basis. Vol. 114 of Mathematics in Science and Engineering. Academic Press, New York (1975). [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. Comput. 83 (2014) 2101–2126. [Google Scholar]
  89. J. Warren, Barycentric coordinates for convex polytopes. Adv. Comput. Math. 6 (1996) 97–108. [CrossRef] [MathSciNet] [Google Scholar]
  90. J. Warren, S. Schaefer, A.N. Hirani and M. Desbrun, Barycentric coordinates for convex sets. Adv. Comput. Math. 27 (2007) 319–338. [CrossRef] [MathSciNet] [Google Scholar]
  91. M. Wicke, M. Botsch and M. Gross, A finite element method on convex polyhedra. Comput. Graphics Forum 26 (2007) 355–364. [CrossRef] [Google Scholar]
  92. O. Zienkiewicz and R. Taylor, The Finite Element Method, 5th edition. Butterworth-Heinemann, London (2000). [Google Scholar]

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