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Supercloseness of Linear DG-FEM and Its Superconvergence Based on the Polynomial Preserving Recovery for Helmholtz Equation

  • Yu DuEmail author
  • Zhimin Zhang
Article
  • 44 Downloads

Abstract

In this paper we study the supercloseness property of the linear discontinuous Galerkin (DG) finite element method and its superconvergence behavior after post-processing by the polynomial preserving recovery (PPR). The error estimate with explicit dependence on the wave number k, the penalty parameter \(\mu \) and the mesh condition parameter \(\alpha \) is derived. We prove the supercloseness between the DG finite element solution and the linear interpolation and the superconvergence for the recovered gradient by the PPR under the assumption \(k(kh)^2\le C_0\) (h is the mesh size) and certain mesh conditions. Furthermore, we estimate the error between the DG numerical gradient and recovered gradient, which motivates us to define the a posteriori error estimator and design a Richardson extrapolation to post-process the recovered gradient by PPR. Finally, some numerical examples are provided to confirm the theoretical results of superconvergence analysis.

Keywords

Helmholtz equation Large wave number Superconvergence The polynomial preserving recovery Discontinuous Galerkin methods 

Notes

References

  1. 1.
    Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aziz, A.K., Kellogg, R.B.: A scattering problem for the Helmholtz equation. Adv. Comput. Methods Partial Diff. Equ.-III 1, 93–95 (1979)MathSciNetGoogle Scholar
  3. 3.
    Babuška, I., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42(3), 451–484 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Babuška, I., Ihlenburg, F., Paik, E.T., Sauter, S.A.: A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Methods Appl. Mech. Eng. 128, 325–359 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part I: grid with superconvergence. SIAM J. Numer. Anal. 41, 2294–2312 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blum, H., Rannacher, R.: Asymptotic error expansion and richardson extrapolation for linear finite elements. Numer. Math. 49, 11–38 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bramble, J .H., Xu, J.: Some estimates for a weighted L\(^2\) projection. Math. Comput. 56(194), 463–476 (1991)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Burman, E., Wu, H., Zhu, L.: Continuous interior penalty finite element method for Helmholtz equation with high wave number: one dimensional analysis. arXiv:1211.1424
  10. 10.
    Chen, Z., Xiang, X.: A source transfer domain decomposition method for Helmholtz equations in unbounded domain. SIAM J. Numer. Anal. 51, 2331–2356 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, L., Xu, J.: Topics on Adaptive Finite Element Methods. In: Tang, T., Xu, J. (eds.) Adaptive Computations: Theory and Algorithms. Science Press, Beijing (2007)Google Scholar
  12. 12.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Pub. Co., New York (1978)zbMATHGoogle Scholar
  13. 13.
    COMSOL AB.: Comsol Multiphysics User’s Guide, 3.5a ed. (2008)Google Scholar
  14. 14.
    Deraemaeker, A., Babuška, I., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Internat. J. Numer. Methods Eng. 46, 471–499 (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Douglas Jr., J., Santos, J.E., Sheen, D.: Approximation of scalar waves in the space–frequency domain. Math. Models Methods Appl. Sci. 4, 509–531 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Du, Y., Wu, H., Zhang, Z.: Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number. arXiv:1703.00156 (2017)
  17. 17.
    Du, Y., Wu, H.: Preasymptotic error analysis of higher order fem and cip-fem for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53(2), 782–804 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Du, Y., Zhu, L.: Preasymptotic error analysis of high order interior penalty discontinuous Galerkin methods for the Helmholtz equation with high wave number. J. Sci. Comput. 67, 130–152 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Engquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Commun. Pure Appl. Math. 32(3), 313–357 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave numbers. SIAM J. Numer. Anal. 47(4), 2872–2896 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Feng, X., Wu, H.: \(hp\)-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80(276), 1997–2024 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, vol. 69. SIAM (2011)Google Scholar
  23. 23.
    Harari, I.: Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Eng. 140(1), 39–58 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Helfrich, P.: Asymptotic expansion for the finite element approximations of parabolic problems. Bonner Math. Schriften 158, 11–30 (1983)MathSciNetGoogle Scholar
  25. 25.
    Huang, Y., Jinchao, X.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comput. 77(263), 1253–1268 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. I. The \(h\)-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. II. The \(h\)-\(p\) version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lakhany, A.M., Marek, I., Whiteman, J.R.: Superconvergence results on mildly structured triangulations. Comput. Methods Appl. Mech. Eng. 189, 1–75 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lin, Q., Zhang, S., Yan, N.: Asymptotic error expansion and defect correction for Sobolev and viscoelasticity type equations. J. Comput. Math. 16, 57–62 (1998)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Marchuk, G., Shaidurov, V.: Difference Methods and Their Extrapolation. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  31. 31.
    Melenk, J.M., Sauter, S.A.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comput. 79(272), 1871–1914 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Melenk, J.M., Sauter, S.A.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49(3), 1210–1243 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Melenk, J.M., Parsania, A., Sauter, S.: General DG-methods for highly indefinite Helmholtz problems. J. Sci. Comput. 57, 536–581 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Naga, A., Zhang, Z.: A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42, 1780–1800 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rivière, B.: Discontinous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  36. 36.
    Schatz, A.H.: An observation concerning Ritz–Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959–962 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, J.: Asymptotic expansions and \(l^\infty \)-error estimates for mixed finite element methods for second order elliptic problems. Numer. Math. 55, 401–430 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wu, H., Zhang, Z.: Can we have superconvergent gradient recovery under adaptive meshes? SIAM J. Numer. Anal. 45, 1701–1722 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139–1152 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Comput. Methods Appl. Mech. Eng. 190, 4289–4299 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang, Z.: Polynomial preserving gradient recovery and a posteriori estimate for bilinear element on irregular quadrilaterals. Int. J. Numer. Anal. Model. 1, 1–24 (2004)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Zhang, Z.: Polynomial preserving recovery for anisotropic and irregular grids. J. Comput. Math. 22, 331–340 (2004)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Zhang, Z., Li, B.: Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numer. Methods Partial Diff. Equ. 15, 151–167 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26, 1192–1213 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zhang, T., Yu, S.: The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method. Appl. Numer. Math. 85, 128–141 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Zhu, L., Du, Y.: Pre-asymptotic error analysis of \(hp\)-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number. Comput. Math. Appl. 70, 917–933 (2015)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zhu, L., Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: \(hp\) version. SIAM J. Numer. Anal. 51(3), 1828–1852 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Beijing Computational Science Research CenterBeijingChina
  2. 2.School of Mathmatics and Computational ScienceXiangtan UniversityXiangtanChina
  3. 3.Department of MathematicsWayne State UniversityDetroitUSA

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