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All issues Volume 47 / No 3 (May-June 2013) ESAIM: M2AN, 47 3 (2013) 689-715 References
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Issue
ESAIM: M2AN
Volume 47, Number 3, May-June 2013
Page(s) 689 - 715
DOI https://doi.org/10.1051/m2an/2012044
Published online 04 March 2013
  1. R. Altmann and C. Carstensen, p1-nonconforming finite elements on triangulations into triangles and quadrilaterals. SIAM J. Numer. Anal. 50 (2012) 418–438. [CrossRef] [MathSciNet] [Google Scholar]
  2. D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337–344. [Google Scholar]
  3. D. N. Arnold and R. Winther, Nonconforming mixed elements for elasticity. Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. Math. Models Methods Appl. Sci. 13 (2003) 295–307. [CrossRef] [Google Scholar]
  4. I. Babuška and M. Suri, Locking effect in the finite element approximation of elasticity problem. Numer. Math. 62 (1992) 439–463. [CrossRef] [MathSciNet] [Google Scholar]
  5. I. Babuška and M. Suri, On locking and robustness in the finie element method. SIAM J. Numer. Anal. 29 (1992) 1261–1293. [CrossRef] [MathSciNet] [Google Scholar]
  6. R. Bank and B. Welfert, A comparison between the mini-element and the Petrov-Galerkin formulations for the generalized Stokes problem. Comput. Methods Appl. Mech. Eng. 83 (1990) 61–68. [CrossRef] [Google Scholar]
  7. 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) 113–124. [CrossRef] [MathSciNet] [Google Scholar]
  8. S. Brenner and L. Scott, The Mathematical Theorey of Finite Element Methods. Springer-Verlag, New York (1994). [Google Scholar]
  9. S.C. Brenner and L.Y. Sung, Linear finite element methods for planar elasticity. Math. Comput. 59 (1992) 321–338. [Google Scholar]
  10. F. Brezzi, M.-O. Bristeau, L.P. Franca, M. Mallet and G. Rogé, A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Meth. Appl. Mech. Eng. 96 (1992) 117–129. [Google Scholar]
  11. F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM-Math. Model. Numer. Anal. 43 (2009) 277–295. [Google Scholar]
  12. F. Brezzi and J. Douglas, Jr. Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225–236. [CrossRef] [MathSciNet] [Google Scholar]
  13. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York. Springer Series Comput. Math. 15 (1991). [CrossRef] [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. (2006) 1872–1896. [Google Scholar]
  15. A.N. Brooks and T.J.R. Hughes, Streamline upwind Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32 (1982) 199–259. [Google Scholar]
  16. Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: A correction. Calcolo 37 (2000) 253–254. [CrossRef] [MathSciNet] [Google Scholar]
  17. Z. Cai, J. Douglas, Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215–232. [CrossRef] [MathSciNet] [Google Scholar]
  18. C. Carstensen and J. Hu, A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107 (2007) 473–502. [CrossRef] [MathSciNet] [Google Scholar]
  19. P.G. Ciarlet, The Finite Element Method for Elliptic Equations. North-Holland, Amsterdam (1978). [Google Scholar]
  20. G.R. Cowper, Gaussian quadrature formulas for triangles. Int. J. Num. Meth. Eng. 7 (1973) 405–408. [CrossRef] [Google Scholar]
  21. M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Math. Model. Anal. Numer. R-3 (1973) 33–75. [Google Scholar]
  22. L.B. da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comp. Phys. 228 (2009) 7215–7232. [Google Scholar]
  23. L.B. 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]
  24. L.B. 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]
  25. J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM Math. Model. Numer. Anal. 33 (1999) 747–770. [Google Scholar]
  26. J. Douglas, Jr. and J. Wang. An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495–508. [Google Scholar]
  27. R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529–550. [CrossRef] [MathSciNet] [Google Scholar]
  28. M. Farhloul and M. Fortin, A mixed nonconforming finite element for the elasticity and Stokes problems. Math. Models Methods Appl. Sci. 9 (1999) 1179–1199. [CrossRef] [Google Scholar]
  29. M. Fortin, A three-dimensional quadratic nonconforming element. Numer. Math. 46 (1985) 269–279. [CrossRef] [MathSciNet] [Google Scholar]
  30. M. Fortin and M. Soulie, A non-conforming piecewise quadratic finite element on the triangle. Int. J. Numer. Meth. Eng. 19 (1983) 505–520. [Google Scholar]
  31. L. Franca, S. Frey and T. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Eng. 95 (1992) 221–242. [Google Scholar]
  32. V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin (1986). [Google Scholar]
  33. V. Gyrya and K. Lipnikov, High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227 (2008) 8841–8854. [CrossRef] [MathSciNet] [Google Scholar]
  34. H. Han, Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984) 223–233. [Google Scholar]
  35. P. Hood and C. Taylor, A numerical solution for the Navier-Stokes equations using the finite element technique. Computers Fluids 1 (1973) 73–100. [Google Scholar]
  36. T.J.R. Hughes and A.N. Brooks, A multidimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows, edited by T.J.R. Hughes. ASME, New York (1979) 19–35. [Google Scholar]
  37. B.M. Irons and A. Razzaque, Experience with the patch test for convergence of finite elements, in The Mathematics of Foundation of the Finite Element Methods with Applications to Partial Differential Equations, edited by A.K. Aziz. Academic Press, New York (1972) 557–587. [Google Scholar]
  38. P. Klouček, B. Li and M. Luskin, Analysis of a class of nonconforming finite elements for crystalline microstructures. Math. Comput. 65 (1996) 1111–1135. [CrossRef] [Google Scholar]
  39. M. Köster, A. Quazzi, F. Schieweck, S. Turek and P. Zajac, New robust nonconforming finite elements of higher order. Appl. Numer. Math. 62 (2012) 166–184. [CrossRef] [Google Scholar]
  40. C.-O. Lee, J. Lee and D. Sheen, A locking-free nonconforming finite element method for planar elasticity. Adv. Comput. Math. 19 (2003) 277–291. [CrossRef] [MathSciNet] [Google Scholar]
  41. H. Lee and D. Sheen, A new quadratic nonconforming finite element on rectangles. Numer. Methods Partial Differ. Equ. 22 (2006) 954–970. [CrossRef] [Google Scholar]
  42. P. Lesaint, On the convergence of Wilson’s nonconforming element for solving the elastic problem. Comput. Methods Appl. Mech. Eng. 7 (1976) 1–76. [Google Scholar]
  43. B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comput. 67 (1998) 917–946. [CrossRef] [MathSciNet] [Google Scholar]
  44. Z.X. Luo, Z.L. Meng and C.M. Liu, Computational Geometry – Theory and Applications of Surface Representation. Sinica Academic Press, Beijing (2010). [Google Scholar]
  45. P. Ming and Z.-C. Shi, Nonconforming rotated Q1 element for Reissner-Mindlin plate. Math. Models Methods Appl. Sci. 11 (2001) 1311–1342. [CrossRef] [Google Scholar]
  46. C. Park and D. Sheen. P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 624–640. [CrossRef] [MathSciNet] [Google Scholar]
  47. R. Pierre, Simple C0 approximations for the computation of incompressible flows. Comput. Methods Appl. Mech. Eng. 68 (1988) 205–227. [CrossRef] [Google Scholar]
  48. R. Pierre, Regularization procedures of mixed finite element approximations of the Stokes problem. Numer. Methods Partial Differ. Equ. 5 (1989) 241–258. [CrossRef] [Google Scholar]
  49. R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8 (1992) 97–111. [Google Scholar]
  50. G. Sander and P. Beckers, The influence of the choice of connectors in the finite element method. Int. J. Numer. Methods Eng. 11 (1977) 1491–1505. [CrossRef] [Google Scholar]
  51. Z.-C. Shi, A convergence condition for the quadrilateral Wilson element. Numer. Math. 44 (1984) 349–361. [CrossRef] [MathSciNet] [Google Scholar]
  52. Z.-C. Shi, On the convergence properties of the quadrilateral elements of Sander and Beckers. Math. Comput. 42 (1984) 493–504. [CrossRef] [Google Scholar]
  53. G. Strang, Variational crimes in the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, edited by A.K. Aziz. New York, Academic Press (1972) 689–710. [Google Scholar]
  54. G. Strang and G.J. Fix, An Analysis of the Finite Element Method. Prentice–Hall, Englewood Cliffs (1973). [Google Scholar]
  55. R. Wang, Multivariate Spline Functions and Their Applications. Science Press, Kluwer Academic Publishers (1994). [Google Scholar]
  56. E. L. Wilson, R. L. Taylor, W. P. Doherty and J. Ghaboussi, Incompatible displacement models, in Numerical and Computer Method in Structural Mechanics, Academic Press, New York (1973) 43–57. [Google Scholar]
  57. Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34 (1997) 640–663. [CrossRef] [MathSciNet] [Google Scholar]

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