Superconvergence of Discontinuous Galerkin Methods for Elliptic Boundary Value Problems (original) (raw)

References

  1. Arnold, D.N., Awanou, G., Winther, R.: Finite elements for symmetric tensors in three dimensions. Math. Comput. 77, 1229–1251 (2008)
    Article MathSciNet Google Scholar
  2. Arnold, D.N. , Brezzi, F.:Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, ESAIM: Mathematical Modelling and Numerical Analysis, 19 (1985), pp. 7–32
  3. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)
    Article MathSciNet Google Scholar
  4. Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numerische Mathematik 92, 401–419 (2002)
    Article MathSciNet Google Scholar
  5. Bank, R.E., Li, Y.: Superconvergent recovery of Raviart-Thomas mixed finite elements on triangular grids. J. Sci. Comput. 81, 1882–1905 (2019)
    Article MathSciNet Google Scholar
  6. Bramble, J.H., Xu, J.: A local post-processing technique for improving the accuracy in mixed finite-element approximations. SIAM J. Numer. Anal. 26, 1267–1275 (1989)
    Article MathSciNet Google Scholar
  7. Brandts, J.H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numerische Mathematik 68, 311–324 (1994)
    Article MathSciNet Google Scholar
  8. Brandts, J.H.: Superconvergence for triangular order k=1 Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods. Appl. Numer. Mathe. 34, 39–58 (2000)
    Article MathSciNet Google Scholar
  9. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, part i: Grids with superconvergence. J. Numer. Anal. 41, 2294–2312 (2003)
    Article MathSciNet Google Scholar
  10. Chen, C., Huang, Y.: High Accuracy Theory of Finite Element Methods, (1995)
  11. Chen, H., Li, B.: Superconvergence analysis and error expansion for the Wilson nonconforming finite element. Numerische Mathematik 69, 125–140 (2013)
    Article MathSciNet Google Scholar
  12. Cockburn, B., Fu, G.: Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions. IMA J. Numer. Anal. 38, 566–604 (2018)
    Article MathSciNet Google Scholar
  13. Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79, 1331–1349 (2010)
    Article MathSciNet Google Scholar
  14. 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. 47, 1319–1365 (2009)
    Article MathSciNet Google Scholar
  15. Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78, 1–24 (2009)
    Article MathSciNet Google Scholar
  16. Cockburn, B., Qiu, W., Shi, K.: Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comput. 81, 1327–1353 (2012)
    Article MathSciNet Google Scholar
  17. Cockburn, B., Shi, K.: Superconvergent HDG methods for linear elasticity with weakly symmetric stresses. IMA J. Numer. Anal. 33, 747–770 (2013)
    Article MathSciNet Google Scholar
  18. Douglas, J., Wang, J.: Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26, 121–133 (1989)
    Article MathSciNet Google Scholar
  19. Gastaldi, L., Nochetto, R.H.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 23 (1989), pp. 103–128
  20. Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32, 352–372 (2012)
    Article MathSciNet Google Scholar
  21. Hong, Q., Hu, J., Ma, L., Xu, J.: An extended Galerkin analysis for linear elasticity with strongly symmetric stress tensor, arXiv preprint arXiv:2002.11664, (2020)
  22. Hong, Q., Hu, J., Shu, S., Xu, J.: A discontinuous Galerkin method for the fourth-order curl problem. J. Comput. Mathe. 565–578 (2012)
  23. Hong, Q., Kraus, J., Xu, J., Zikatanov, L.: A robust multigrid method for discontinuous Galerkin discretizations of stokes and linear elasticity equations. Numerische Mathematik 132, 23–49 (2016)
    Article MathSciNet Google Scholar
  24. Hong, Q., Wang, F., Wu, S., Xu, J.: A unified study of continuous and discontinuous Galerkin methods. Sci. China Math. 62, 1–32 (2019)
    Article MathSciNet Google Scholar
  25. Hong, Q., Wu, S., Xu, J.: An Extended Galerkin Analysis for Elliptic Problems, arXiv preprint arXiv:1908.08205v2, (2019)
  26. Hong, Q., Xu, J.: Uniform stability and error analysis for some discontinuous galerkin methods, arXiv preprint arXiv:1805.09670, (2018)
  27. Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the higher order case. J. Comput. Math. 33, 1–14 (2015)
    Article MathSciNet Google Scholar
  28. Hu, J., Ma, L., Ma, R.: Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements, arXiv preprint arXiv:1808.09810, (2018)
  29. Hu, J., Ma, R.: Superconvergence of both the Crouzeix-Raviart and Morley elements. Numerische Mathematik 132, 491–509 (2016)
    Article MathSciNet Google Scholar
  30. Hu, J., Zhang, S.: A family of conforming mixed finite elements for linear elasticity on triangular grids, arXiv preprint arXiv:1406.7457, (2014)
  31. Hu, J., Zhang, S.: A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math. 58, 297–307 (2015)
    Article MathSciNet Google Scholar
  32. Hu, J., Zhang, S.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): The lower order case. Math. Models Methods Appl. Sci. 26, 1649–1669 (2016)
    Article MathSciNet Google Scholar
  33. Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numerische Mathematik 53, 513–538 (1988)
    Article MathSciNet Google Scholar
  34. Stenberg, R.: Postprocessing schemes for some mixed finite elements, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 25 (1991), pp. 151–167
  35. Wang, F., Wu, S., Xu, J.:A mixed discontinuous Galerkin method for linear elasticity with strongly imposed symmetry, arXiv preprint arXiv:1902.08717, (2019)
  36. Xie, Z., Zhang, Z., Zhang, Z.: A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems. J. Comput. Math. 280–298 (2009)
  37. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581–613 (1992)
    Article MathSciNet Google Scholar

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