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Science China Mathematics

, Volume 62, Issue 1, pp 1–32 | Cite as

A unified study of continuous and discontinuous Galerkin methods

  • Qingguo Hong
  • Fei Wang
  • Shuonan Wu
  • Jinchao XuEmail author
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Abstract

A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

Keywords

finite element methods DG-derivatives unified study 

MSC(2010)

65N30 65M60 65M12 

Notes

Acknowledgements

The last author was supported by US Department of Energy (Grant No. DE-SC-0009249), as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials, US Department of Energy (Grant No. DE-SC0014400) and National Science Foundation of USA (Grant No. DMS-1522615). The second author was partially supported by National Natural Science Foundation of China (Grant No. 11771350).

References

  1. 1.
    Argyris J H. Energy theorems and structural analysis: A generalized discourse with applications on energy principles of structural analysis including the effects of temperature and non-linear stress-strain relations. Aircraft Eng Aerosp Tech, 1954, 26: 347–356Google Scholar
  2. 2.
    Arnold D N. An interior penalty finite element method with discontinuous elements. SIAM J Numer Anal, 1982, 19: 742–760MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arnold D N, Brezzi F. Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Modél Math Anal Numér, 1985, 19: 7–32MathSciNetzbMATHGoogle Scholar
  4. 4.
    Arnold D N, Brezzi F, Cockburn B, et al. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal, 2002, 39: 1749–1779MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aubin J P. Approximation des problemes aux limites non homogenes pour des opérateurs non linéaires. J Math Anal Appl, 1970, 30: 510–521MathSciNetzbMATHGoogle Scholar
  6. 6.
    Babuška I, Oden J T, Lee J K. Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems. Comput Methods Appl Mech Engrg, 1977, 11: 175–206MathSciNetzbMATHGoogle Scholar
  7. 7.
    Babuška I, Zlámal M. Nonconforming elements in the finite element method with penalty. SIAM J Numer Anal, 1973, 10: 863–875MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bassi F, Rebay S, Mariotti G, et al. A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics. Antwerpen: Technologisch Instituut, 1997, 99–109Google Scholar
  9. 9.
    Becker R, Burman E, Hansbo P, et al. A reduced P1-discontinuous Galerkin method. Göteborg: Chalmers Finite Element Center, https://fenicsproject.org/pub/femcenter/pub/preprints/pdf/phiprint-2003-13.pdf, 2004Google Scholar
  10. 10.
    Beir~ao da Veiga L, Brezzi F, Cangiani A, et al. Basic principles of virtual element methods. Math Models Methods Appl Sci, 2013, 23: 199–214MathSciNetzbMATHGoogle Scholar
  11. 11.
    Boff D, Brezzi F, Fortin M. Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Berlin: Springer, 2013Google Scholar
  12. 12.
    Brenner S C. Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J Numer Anal, 2003, 41: 306–324MathSciNetzbMATHGoogle Scholar
  13. 13.
    Brenner S C. Forty years of the Crouzeix-Raviart element. In: Numer Methods Partial Differential Equations, vol. 31. New York: Wiley, 2015, 367–396MathSciNetzbMATHGoogle Scholar
  14. 14.
    Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. Berlin: Springer, 2007Google Scholar
  15. 15.
    Brenner S C, Sung L-Y. C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J Sci Comput, 2005, 22: 83–118MathSciNetzbMATHGoogle Scholar
  16. 16.
    Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Anal Numér, 1974, 8: 129–151MathSciNetzbMATHGoogle Scholar
  17. 17.
    Brezzi F, Douglas Jr J, Durán R, et al. Mixed finite elements for second order elliptic problems in three variables. Numer Math, 1987, 51: 237–250MathSciNetzbMATHGoogle Scholar
  18. 18.
    Brezzi F, Douglas Jr J, Fortin M, et al. Effcient rectangular mixed finite elements in two and three space variables. RAIRO Modél Math Anal Numér, 1987, 21: 581–604MathSciNetGoogle Scholar
  19. 19.
    Brezzi F, Douglas Jr J, Marini L D. Two families of mixed finite elements for second order elliptic problems. Numer Math, 1985, 47: 217–235MathSciNetzbMATHGoogle Scholar
  20. 20.
    Brezzi F, Falk R S, Marini L D. Basic principles of mixed virtual element methods. ESAIM Math Model Numer Anal, 2014, 48: 1227–1240MathSciNetzbMATHGoogle Scholar
  21. 21.
    Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Berlin: Springer-Verlag, 1991Google Scholar
  22. 22.
    Brezzi F, Hughes T, Marini L D, et al. Mixed discontinuous Galerkin methods for Darcy flow. J Sci Comput, 2005, 22: 119–145MathSciNetzbMATHGoogle Scholar
  23. 23.
    Brezzi F, Lipnikov K, Shashkov M. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J Numer Anal, 2005, 43: 1872–1896MathSciNetzbMATHGoogle Scholar
  24. 24.
    Brezzi F, Lipnikov K, Simoncini V. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math Models Methods Appl Sci, 2005, 15: 1533–1551MathSciNetzbMATHGoogle Scholar
  25. 25.
    Brezzi F, Manzini G, Marini D, et al. Discontinuous finite elements for diffusion problems. Milano: Atti Convegno in Onore di F Brioschi, 1999, 197–217Google Scholar
  26. 26.
    Brezzi F, Manzini G, Marini L D, et al. Discontinuous Galerkin approximations for elliptic problems. Numer Methods Partial Differential Equations, 2000, 16: 365–378MathSciNetzbMATHGoogle Scholar
  27. 27.
    Burman E, Stamm B. Local discontinuous Galerkin method for diffusion equations with reduced stabilization. Commun Comput Phys, 2009, 5: 498–514MathSciNetzbMATHGoogle Scholar
  28. 28.
    Carrero J, Cockburn B, Schötzau D. Hybridized globally divergence-free LDG methods. Part I: The Stokes problem. Math Comp, 2006, 75: 533–563MathSciNetzbMATHGoogle Scholar
  29. 29.
    Céa J. Approximation variationnelle des problèmes aux limites. Ann Inst Fourier (Grenoble), 1964, 14: 345–444MathSciNetzbMATHGoogle Scholar
  30. 30.
    Chen W, Wang F, Wang Y. Weak Galerkin method for the coupled Darcy-Stokes flow. IMA J Numer Anal, 2016, 36: 897–921MathSciNetzbMATHGoogle Scholar
  31. 31.
    Chen Y, Cockburn B. Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: General nonconforming meshes. IMA J Numer Anal, 2012, 32: 1267–1293zbMATHGoogle Scholar
  32. 32.
    Chen Y, Cockburn B. Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes. Math Comp, 2014, 83: 87–111zbMATHGoogle Scholar
  33. 33.
    Chen Y, Huang J, Huang X, et al. On the local discontinuous Galerkin method for linear elasticity. Math Prob Eng, 2010, 2010: 759547MathSciNetzbMATHGoogle Scholar
  34. 34.
    Chung E, Cockburn B, Fu G. The staggered DG method is the limit of a hybridizable DG method. SIAM J Numer Anal, 2014, 52: 915–932MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ciarlet P G. The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. Amsterdam: North-Holland, 1978Google Scholar
  36. 36.
    Ciarlet P G, Wagschal C. Multipoint Taylor formulas and applications to the finite element method. Numer Math, 1971, 17: 84–100MathSciNetzbMATHGoogle Scholar
  37. 37.
    Cockburn B. Static condensation, hybridization, and the devising of the HDG methods. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Berlin: Springer, 2016, 129–177Google Scholar
  38. 38.
    Cockburn B, Dong B, Guzmán J. A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math Comp, 2008, 77: 1887–1916MathSciNetzbMATHGoogle Scholar
  39. 39.
    Cockburn B, Fu G. Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions. IMA J Numer Anal, 2017, doi: 10.1093/imanum/drx025Google Scholar
  40. 40.
    Cockburn B, Fu G, Sayas F. Superconvergence by M-decompositions. Part I: General theory for HDG methods for diffusion. Math Comp, 2017, 86: 1609–1641zbMATHGoogle Scholar
  41. 41.
    Cockburn B, Gopalakrishnan J. A characterization of hybridized mixed methods for second order elliptic problems. SIAM J Numer Anal, 2004, 42: 283–301MathSciNetzbMATHGoogle Scholar
  42. 42.
    Cockburn B, Gopalakrishnan J. Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math Comp, 2005, 74: 1653–1677MathSciNetzbMATHGoogle Scholar
  43. 43.
    Cockburn B, Gopalakrishnan J. Incompressible finite elements via hybridization. Part I: The Stokes system in two space dimensions. SIAM J Numer Anal, 2005, 43: 1627–1650zbMATHGoogle Scholar
  44. 44.
    Cockburn B, Gopalakrishnan J. Incompressible finite elements via hybridization. Part II: The Stokes system in three space dimensions. SIAM J Numer Anal, 2005, 43: 1651–1672zbMATHGoogle Scholar
  45. 45.
    Cockburn B, Gopalakrishnan J. New hybridization techniques. GAMM-Mitt, 2005, 28: 154–182MathSciNetzbMATHGoogle Scholar
  46. 46.
    Cockburn B, Gopalakrishnan J, Lazarov R. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J Numer Anal, 2009, 47: 1319–1365MathSciNetzbMATHGoogle Scholar
  47. 47.
    Cockburn B, Gopalakrishnan J, Sayas F. A projection-based error analysis of HDG methods. Math Comp, 2010, 79: 1351–1367MathSciNetzbMATHGoogle Scholar
  48. 48.
    Cockburn B, Karniadakis G E, Shu C-W. The development of discontinuous Galerkin methods. In: Discontinuous Galerkin Methods. Berlin: Springer, 2000, 3–50Google Scholar
  49. 49.
    Cockburn B, Mustapha K. A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer Math, 2015, 130: 293–314MathSciNetzbMATHGoogle Scholar
  50. 50.
    Cockburn B, Nochetto R H, Zhang W. Contraction property of adaptive hybridizable discontinuous Galerkin methods. Math Comp, 2016, 85: 1113–1141MathSciNetzbMATHGoogle Scholar
  51. 51.
    Cockburn B, Qiu W, Shi K. Conditions for superconvergence of HDG methods for second-order elliptic problems. Math Comp, 2012, 81: 1327–1353MathSciNetzbMATHGoogle Scholar
  52. 52.
    Cockburn B, Qiu W, Shi K. Superconvergent HDG methods on isoparametric elements for second-order elliptic problems. SIAM J Numer Anal, 2012, 50: 1417–1432MathSciNetzbMATHGoogle Scholar
  53. 53.
    Cockburn B, Shen J. A hybridizable discontinuous Galerkin method for the p-Laplacian. SIAM J Sci Comput, 2016, 38: 545–566MathSciNetzbMATHGoogle Scholar
  54. 54.
    Cockburn B, Shu C-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal, 1998, 35: 2440–2463MathSciNetzbMATHGoogle Scholar
  55. 55.
    Cockburn B, Zhang W. A posteriori error estimates for HDG methods. J Sci Comput, 2012, 51: 582–607MathSciNetzbMATHGoogle Scholar
  56. 56.
    Cockburn B, Zhang W. A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems. SIAM J Numer Anal, 2013, 51: 676–693MathSciNetzbMATHGoogle Scholar
  57. 57.
    Courant R. Variational methods for the solution of problems of equilibrium and vibrations. Bull Amer Math Soc (NS), 1943, 49: 1–23MathSciNetzbMATHGoogle Scholar
  58. 58.
    Crouzeix M, Raviart P A. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Rev Française Automat Informat Recherche Opérationnelle Sér Rouge, 1973, 7: 33–75MathSciNetzbMATHGoogle Scholar
  59. 59.
    Douglas Jr J, Dupont T. Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing Methods in Applied Sciences. Berlin: Springer, 1976, 207–216Google Scholar
  60. 60.
    Ekeland I, Temam R. Convex Analysis and Variational Problems. Philadelphia: SIAM, 1976zbMATHGoogle Scholar
  61. 61.
    Eymard R, Gallouët T, Herbin R. Finite volume methods. In: Handbook of Numerical Analysis, vol. 7. New York: Elsevier, 2000, 713–1018MathSciNetzbMATHGoogle Scholar
  62. 62.
    Feng K. Finite difference method based on variation principle. Commun Appl Math Comput, 1965, 2: 237–261Google Scholar
  63. 63.
    Fortin M. A three-dimensional quadratic nonconforming element. Numer Math, 1985, 46: 269–279MathSciNetzbMATHGoogle Scholar
  64. 64.
    Fortin M, Soulie M. A non-conforming piecewise quadratic finite element on triangles. Internat J Numer Methods Engrg, 1983, 19: 505–520MathSciNetzbMATHGoogle Scholar
  65. 65.
    Fraeijs de Veubeke B. Displacement and equilibrium models in the finite element method. In: Stress Analysis. New York: Wiley, 1965, 145–197Google Scholar
  66. 66.
    Fu G, Cockburn B, Stolarski H. Analysis of an HDG method for linear elasticity. Internat J Numer Methods Engrg, 2015, 102: 551–575MathSciNetzbMATHGoogle Scholar
  67. 67.
    Gong S, Wu S, Xu J. New hybridized mixed methods for linear elasticity and optimal multilevel solvers. ArXiv: 1704.07540, 2017Google Scholar
  68. 68.
    Henshell R D. On hybrid finite elements. In: The Mathematics of Finite Elements and Applications. London-New York: Academic Press, 1973, 299–312Google Scholar
  69. 69.
    Hong Q, Hu J, Shu S, et al. A discontinuous Galerkin method for the fourth-order curl problem. J Comput Math, 2012, 30: 565–578MathSciNetzbMATHGoogle Scholar
  70. 70.
    Hong Q, Kraus J. Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem. SIAM J Numer Anal, 2016, 54: 2750–2774MathSciNetzbMATHGoogle Scholar
  71. 71.
    Hong Q, Kraus J. Parameter-robust stability of classical three-field formulation of Biot's consolidation model. ArXiv: 1706.00724, 2017zbMATHGoogle Scholar
  72. 72.
    Hong Q, Kraus J, Xu J, et al. A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations. Numer Math, 2016, 132: 23–49MathSciNetzbMATHGoogle Scholar
  73. 73.
    Hong Q, Wu S, Xu J. Extended Galerkin methods for second order elliptic problems. Preprint, 2018Google Scholar
  74. 74.
    Hong Q, Xu J. Uniform stability and error analysis for some discontinuous Galerkin methods. ArXiv:1805.09670, 2018Google Scholar
  75. 75.
    Hrennikoff A. Solution of problems of elasticity by the framework method. J Appl Mech, 1941, 8: 169–175MathSciNetzbMATHGoogle Scholar
  76. 76.
    Hu J, Zhang S. A canonical construction of H m-nonconforming triangular finite elements. Ann Appl Math, 2017, 3: 266–288zbMATHGoogle Scholar
  77. 77.
    Jones R E. A generalization of the direct-stiffness method of structural analysis. AIAA J, 1964, 2: 821–826zbMATHGoogle Scholar
  78. 78.
    Kabaria H, Lew A J, Cockburn B. A hybridizable discontinuous Galerkin formulation for non-linear elasticity. Comput Methods Appl Mech Engrg, 2015, 283: 303–329MathSciNetzbMATHGoogle Scholar
  79. 79.
    Lions J L. Problèms aux limites non homogènes à donées irrégulières: Une méthode d'approximation. In: Numerical Analysis of Partial Differential Equations. Rome: Edizioni Cremonese, 1968, 283–292Google Scholar
  80. 80.
    Liu H, Yan J. The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J Numer Anal, 2009, 47: 675–698MathSciNetzbMATHGoogle Scholar
  81. 81.
    Liu H, Yan J. The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun Comput Phys, 2010, 8: 541–564MathSciNetzbMATHGoogle Scholar
  82. 82.
    Mikhlin S G. Variational Methods in Mathematical Physics. New York: Macmillan, 1964zbMATHGoogle Scholar
  83. 83.
    Mu L, Wang J, Ye X. Weak Galerkin finite element methods on polytopal meshes. Int J Numer Anal Model, 2015, 12: 31–53MathSciNetzbMATHGoogle Scholar
  84. 84.
    Mu L, Wang J, Ye X, et al. A weak Galerkin finite element method for the Maxwell equations. J Sci Comput, 2015, 65: 363–386MathSciNetzbMATHGoogle Scholar
  85. 85.
    Nicolaides R A. On a class of finite elements generated by Lagrange interpolation. SIAM J Numer Anal, 1972, 9: 435–445MathSciNetzbMATHGoogle Scholar
  86. 86.
    Nédélec J C. Mixed finite elements in R3. Numer Math, 1980, 35: 315–341MathSciNetzbMATHGoogle Scholar
  87. 87.
    Nédélec J C. A new family of mixed finite elements in R3. Numer Math, 1986, 50: 57–81MathSciNetzbMATHGoogle Scholar
  88. 88.
    Oikawa I. A hybridized discontinuous Galerkin method with reduced stabilization. J Sci Comput, 2015, 65: 327–340MathSciNetzbMATHGoogle Scholar
  89. 89.
    Park C, Sheen D. P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J Numer Anal, 2003, 41: 624–640MathSciNetzbMATHGoogle Scholar
  90. 90.
    Pian T H H. Derivation of element stiffness matrices by assumed stress distributions. AIAA J, 1964, 2: 1333–1336Google Scholar
  91. 91.
    Pian T H H. Finite element formulation by variational principles with relaxed continuity requirements. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Tuscaloosa: Academic Press, 1972, 671–687Google Scholar
  92. 92.
    Pian T H H, Tong P. Basis of finite element methods for solid continua. Internat J Numer Methods Engrg, 1969, 1: 3–28zbMATHGoogle Scholar
  93. 93.
    Pin T, Pian T H H. A variational principle and the convergence of a finite-element method based on assumed stress distribution. Internat J Solids Structures, 1969, 5: 463–472zbMATHGoogle Scholar
  94. 94.
    Raviart P A. Hybrid finite element methods for solving 2nd order elliptic equations. In: Topics in Numerical Analysis, II. London: Academic Press, 1975, 141–155Google Scholar
  95. 95.
    Raviart P A, Thomas J M. A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Berlin: Springer, 1977, 292–315Google Scholar
  96. 96.
    Raviart P A, Thomas J M. Primal hybrid finite element methods for 2nd order elliptic equations. Math Comp, 1977, 31: 391–413MathSciNetzbMATHGoogle Scholar
  97. 97.
    Roberts J E, Thomas J M. Mixed and hybrid methods. In: Handbook of Numerical Analysis, vol. 2. New York: Elsevier, 1991, 523–639MathSciNetzbMATHGoogle Scholar
  98. 98.
    Santos J E, Sheen D, Ye X. Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. M2AN Math Model Numer Anal, 1999, 33: 747–770MathSciNetzbMATHGoogle Scholar
  99. 99.
    Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp, 1990, 54: 483–493MathSciNetzbMATHGoogle Scholar
  100. 100.
    Soon S-C, Cockburn B, Stolarski H K. A hybridizable discontinuous Galerkin method for linear elasticity. Internat J Numer Methods Engrg, 2009, 80: 1058–1092MathSciNetzbMATHGoogle Scholar
  101. 101.
    Strang G. Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. New York: Academic Press, 1972, 689–710Google Scholar
  102. 102.
    Sun S, Liu J. A locally conservative finite element method based on piecewise constant enrichment of the continuous Galerkin method. SIAM J Sci Comput, 2009, 31: 2528–2548MathSciNetzbMATHGoogle Scholar
  103. 103.
    Turner M J, Clough R W, Martin H C, et al. Stiffness and de ection analysis of complex structures. J Aero Sci, 1956, 23: 805–823zbMATHGoogle Scholar
  104. 104.
    Veeser A, Zanotti P. Quasi-optimal nonconforming methods for symmetric elliptic problems, I—Abstract theory. ArXiv:1710.03331, 2017zbMATHGoogle Scholar
  105. 105.
    Veeser A, Zanotti P. Quasi-optimal nonconforming methods for symmetric elliptic problems, III—DG and other interior penalty methods. ArXiv:1710.03452, 2017zbMATHGoogle Scholar
  106. 106.
    Wang C, Wang J. A primal-dual weak Galerkin finite element method for Fokker-Planck type equations. ArXiv: 1704.05606, 2017Google Scholar
  107. 107.
    Wang C, Wang J. A primal-dual weak Galerkin finite element method for second order elliptic equations in nondivergence form. Math Comp, 2018, 87: 515–545MathSciNetzbMATHGoogle Scholar
  108. 108.
    Wang J, Wang C. Weak Galerkin finite element methods for elliptic PDEs (in Chinese). Sci Sin Math, 2015, 45: 1061–1092Google Scholar
  109. 109.
    Wang J, Ye X. A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math, 2013, 241, 103–115MathSciNetzbMATHGoogle Scholar
  110. 110.
    Wang J, Ye X. A weak Galerkin mixed finite element method for second order elliptic problems. Math Comp, 2014, 83: 2101–2126MathSciNetzbMATHGoogle Scholar
  111. 111.
    Wang J, Ye X. A weak Galerkin finite element method for the Stokes equations. Adv Comput Math, 2016, 42: 155–174MathSciNetzbMATHGoogle Scholar
  112. 112.
    Wang M, Xu J. The Morley element for fourth order elliptic equations in any dimensions. Numer Math, 2006, 103: 155–169MathSciNetzbMATHGoogle Scholar
  113. 113.
    Wang M, Xu J. Minimal finite element spaces for 2m-th-order partial differential equations in Rn. Math Comp, 2013, 82: 25–43MathSciNetzbMATHGoogle Scholar
  114. 114.
    Wheeler M F. An elliptic collocation-finite element method with interior penalties. SIAM J Numer Anal, 1978, 15: 152–161MathSciNetzbMATHGoogle Scholar
  115. 115.
    Wilson E L, Taylor R L, Doherty W P, et al. Incompatible displacement models. In: Proceedings of the Symposium on Numerical and Computer Methods in Structural Engineering. New York: Academic Press, 1973, 43–57Google Scholar
  116. 116.
    Wolf J P. Alternate hybrid stress finite element models. Internat J Numer Methods Engrg, 1975, 9: 601–615zbMATHGoogle Scholar
  117. 117.
    Wu S, Gong S, Xu J. Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor. Math Models Methods Appl Sci, 2017, 27: 2711–2743MathSciNetzbMATHGoogle Scholar
  118. 118.
    Wu S, Xu J. P m interior penalty nonconforming finite element methods for 2m-th order PDEs in Rn. ArXiv: 1710.07678, 2017Google Scholar
  119. 119.
    Wu S, Xu J. Nonconforming finite element spaces for 2m-th order partial differential equations on Rn simplicial grids when m = n + 1. Math Comp, 2018, in pressGoogle Scholar
  120. 120.
    Yamamoto Y. A formulation of matrix displacement method. PhD Thesis. Cambridge: Massachusetts Institute of Technology, 1966Google Scholar
  121. 121.
    Zlámal M. On the finite element method. Numer Math, 1968, 12: 394–409MathSciNetzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Qingguo Hong
    • 1
  • Fei Wang
    • 2
  • Shuonan Wu
    • 1
  • Jinchao Xu
    • 1
    Email author
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.School of Mathematics and StatisticsXi′an Jiaotong UniversityXi′anChina

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