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Polynomial Preserving Recovery for High Frequency Wave Propagation

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Abstract

Polynomial preserving recovery (PPR) was first proposed and analyzed in Zhang and Naga in SIAM J Sci Comput 26(4):1192–1213, (2005), with intensive following applications on elliptic problems. In this paper, we generalize the study of PPR to high-frequency wave propagation. Specifically, we establish the supercloseness between finite element solution and its interpolation with explicit dependence on the frequency of wavefield, and then prove the superconvergence of PPR for high-frequency solutions to wave equation based on the supercloseness. We also present several numerical examples of PPR for both low-frequency and high-frequency wave propagation in order to confirm the theoretical results of superconvergence analysis.

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References

  1. Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  2. Babuška, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K., Copps, K.: Validation of a posteriori error estimators by numerical approach. Int. J. Numer. Methods Eng. 37(7), 1073–1123 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baccouch, M.: A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids. Comput. Math. Appl. 68(10), 1250–1278 (2014)

    Article  MathSciNet  Google Scholar 

  4. Baker, G.A.: Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal. 13(4), 564–576 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bank, R., Xu, J.: Asymptotically exact a posteriori error estimators. I. Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods, 3rd edn. In: Texts in Applied Mathematics, vol. 15. Springer, New York (2008)

  7. Carstensen, C., Bartels, S.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM. Math. Comput. 71(239), 945–969 (2002). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, C., Hu, S.: The highest order superconvergence for bi-k degree rectangular elements at nodes: a proof of 2k-conjecture. Math. Comput. 82(283), 1337–1355 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, L., Xu, J.: A posteriori error estimator by post-processing. In: Tang, T., Xu, J. (eds.) Adaptive Computations: Theory and Algorithms. Science Press, Beijing (2006)

    Google Scholar 

  10. Chou, C.-S., Shu, C.-W., Xing, Y.: Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272, 88–107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  12. Cockburn, B., Quenneville-Bélair, V.: Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation. Math. Comput. 83(285), 65–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Douglas, J. Jr., Dupont, T.: Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems. In: Topics in Numerical Analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972), pp. 89–92. Academic Press, London (1973)

  14. Douglas, J., Dupont, T.: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22, 99–109 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dougalis, V.A., Serbin, S.M.: On the superconvergence of Galerkin approximations to second-order hyperbolic equations. SIAM J. Numer. Anal. 17(3), 431–446 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dupont, T.: \(L^2\)-estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10, 880–889 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  18. Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Formaggia, L., Perotto, S.: Anisotropic error estimates for elliptic problems. Numer. Math. 94(1), 67–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo, H., Zhang, Z.: Gradient recovery for the Crouzeix–Raviart element. J. Sci. Comput. 64(2), 456–476 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, H., Zhang, Z., Zhao, R.: Superconvergent Two-grid Methods For Elliptic Eigenvalue Problems, J Sci Comput, 1–24 , (2016). doi:10.1007/s10915-016-0245-2

  22. Huang, W., Russell, R.: Adaptive Moving Mesh Methods, Applied Mathematical Sciences, 174. Springer, New York (2011)

    Book  Google Scholar 

  23. Huang, Y., Xu, J.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comput. 77(263), 1253–1268 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang, Z.Y., Yang, X.: Tailored finite point method for first order wave equation. J. Sci. Comput. 49(3), 351–366 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lakhany, A.M., Marek, I., Whiteman, J.R.: Superconvergence results on mildly structured triangulations. Comput. Methods Appl. Mech. Eng. 189(1), 1–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ihlenburg, F., Babus̆ka, 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ihlenburg, F., Babus̆ka, 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lin, Q., Wang, H., Lin, T.: Interpolated finite element methods for second order hyperbolic equations and their global superconvergence. Syst. Sci. Math. Sci. 6(4), 331–340 (1993)

    MathSciNet  MATH  Google Scholar 

  29. Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. I. Springer, New York (1972)

    Book  MATH  Google Scholar 

  30. Naga, A., Zhang, Z.: A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42(4), 1780–1800 (2004). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  31. Naga, A., Zhang, Z.: The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete Contin. Dyn. Syst. Ser. B 5(3), 769–798 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Naga, A., Zhang, Z., Zhou, A.: Enhancing eigenvalue approximation by gradient recovery. SIAM J. Sci. Comput. 28(4), 1289–1300 (2006). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Niceno, EasyMesh Version 1.4: A Two-Dimensional Quality Mesh Generator, http://www-dinma.univ.trieste.it/nirftc/research/easymesh

  34. Shi, D., Li, Z.: Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. Appl. Math. J. Chin. Univ. Ser. B 23(4), 455–462 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sjögreen, B., Petersson, N.A.: A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation. J. Sci. Comput. 52(1), 17–48 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wahlbin, L.: Superconvergence in Galerkin finite element methods. Lecture Notes in Mathematics, vol. 1605. Springer, Berlin (1995)

  37. Wang, F., Chen, Y., Tang, Y.: Superconvergence of fully discrete splitting positive definite mixed FEM for hyperbolic equations. Numer. Methods Partial Differ. Equ. 30(1), 175–186 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version. IMA J. Numer. Anal. 34(3), 1266–1288 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wu, H., Zhang, Z.: Can we have superconvergent gradient recovery under adaptive meshes? SIAM J. Numer. Anal. 45(4), 1701–1722 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, H., Zhang, Z.: Enhancing eigenvalue approximation by gradient recovery on adaptive meshes. IMA J. Numer. Anal. 29(4), 1008–1022 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xing, Y., Chou, C.-S., Shu, C.-W.: Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Probl. Imaging 7(3), 967–986 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73(247), 1139–1152 (2004). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lu, J., Yang, X.: Frozen Gaussian approximation for high frequency wave propagation. Commun. Math. Sci. 9(3), 663–683 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26(4), 1192–1213 (2005). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24, 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique. Int. J. Numer. Methods Eng. 33, 1331–1364 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. II. Error estimates and adaptivity. Int. J. Numer. Methods Eng. 33, 1331–1364 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Hailong Guo & Xu Yang

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  1. Hailong Guo
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  2. Xu Yang
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Correspondence to Hailong Guo.

Additional information

This work was partially supported by the NSF Grants DMS-1418936 and DMS-1107291, and Hellman Family Foundation Faculty Fellowship, UC Santa Barbara. We also acknowledge support from the Center for Scientific Computing from the CNSI, MRL: an NSF MRSEC (DMR-1121053) and NSF CNS-0960316.

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Guo, H., Yang, X. Polynomial Preserving Recovery for High Frequency Wave Propagation. J Sci Comput 71, 594–614 (2017). https://doi.org/10.1007/s10915-016-0312-8

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