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Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Methods

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Abstract

In this paper, using Fourier analysis technique, we study the super convergence property of the DDGIC (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and the symmetric DDG (Vidden and Yan in J Comput Math 31(6):638–662, 2013) methods for diffusion equation. With \(k\hbox {th}\) degree piecewise polynomials applied, the convergence to the solution’s spatial derivative is \(k\hbox {th}\) order measured under regular norms. On the other hand when measuring the error in the weak sense or in its moment format, the error is super convergent with \((k+2)\hbox {th}\) and \((k+3)\hbox {th}\) orders for its first two moments with even order degree polynomial approximations. We carry out Fourier type (Von Neumann) error analysis and obtain the desired super convergent orders for the case of \(P^2\) quadratic polynomial approximations. The theoretical predicted errors agree well with the numerical results.

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References

  1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baumann, C.E., Oden, J.T.: A discontinuous \(hp\) finite element method for convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cao, W.-X., Liu, H., Zhang, Z.-M.: Superconvergence of the direct discontinuous Galerkin method for convection–diffusion equations. Numer. Methods Partial Differ. Equ. 33(1), 290–317 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, Z., Huang, H., Yan, J.: Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes. J. Comput. Phys. 308, 198–217 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng, Y., Shu, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227, 9612–9627 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cheng, Y., Shu, C.-W.: Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations. Comput. Struct. 87, 630–641 (2009)

    Article  Google Scholar 

  7. Epshteyn, Y.: Discontinuous Galerkin methods for the chemotaxis and haptotaxis models. J. Comput. Appl. Math. 224(1), 168–181 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin methods for the Keller–Segel chemotaxis model. SIAM J. Numer. Anal. 47(1), 386–408 (2008/2009)

  9. Guo, W., Zhong, X., Qiu, J.-M.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach. J. Comput. Phys. 235, 458–485 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang, H., Chen, Z., Li, J., Yan, J.: Direct discontinuous Galerkin method and its variations for second order elliptic equations. J. Sci. Comput. 70(2), 744–765 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  11. Huang, H., Zhong, X., Yan, J.: Direct discontinuous galerkin with interface correction and symmetric direct discontinuous galerkin methods for Keller–Segel chemotaxis equation. SIAM J. Numer. Anal. to be submitted

  12. Li, X.H., Shu, C.-W., Yang, Y.: Local discontinuous Galerkin methods for the Keller–Segel chemotaxis model. J. Sci. Comput. (2017). doi:10.1007/s10915-016-0354-y

  13. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 475–698 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)

    MATH  MathSciNet  Google Scholar 

  15. Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902–931 (2001). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  16. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys. 83(1), 32–78 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  18. Tadmor, E., Liu, Y.-J., Shu, C.-W., Zhang, M.: \({L}^2\) stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM Math. Model. Numer. Anal. 42, 593–607 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Tadmor, E., Liu, Y.-J., Shu, C.-W., Zhang, M.: Central local discontinuous Galerkin methods on overlapping cells for diffusion equations. ESAIM Math. Model. Numer. Anal. 45, 1009–1032 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Vidden, C., Yan, J.: A new direct discontinuous Galerkin method with symmetric structure for nonlinear diffusion equations. J. Comput. Math. 31(6), 638–662 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yan, J.: A new nonsymmetric discontinuous Galerkin method for time dependent convection diffusion equations. J. Sci. Comput. 54(2–3), 663–683 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhang, M., Shu, C.-W.: An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13(3), 395–413 (2003). Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday

  23. Zhang, M., Shu, C.-W.: An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids 23, 581–592 (2005)

    Article  MATH  Google Scholar 

  24. Zhang, M., Yan, J.: Fourier type error analysis of the direct discontinuous Galerkin method and its variations for diffusion equations. J. Sci. Comput. 52(3), 638–655 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zhong, X., Shu, C.-W.: Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng. 200(41–44), 2814–2827 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Mengping Zhang

  2. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Jue Yan

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  1. Mengping Zhang
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  2. Jue Yan
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Correspondence to Jue Yan.

Additional information

Mengping Zhang: The research of this author is supported by NSFC Grant 11471305. Jue Yan: The research of this author is supported by NSF Grant DMS-1620335.

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Zhang, M., Yan, J. Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Methods. J Sci Comput 73, 1276–1289 (2017). https://doi.org/10.1007/s10915-017-0438-3

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  • DOI: https://doi.org/10.1007/s10915-017-0438-3

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