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Superconvergence of immersed finite element methods for interface problems

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Abstract

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.

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References

  1. Adjerid, S., Lin, T.: Higher-order immersed discontinuous Galerkin methods. Int. J. Inf. Syst. Sci. 3(4), 555–568 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Adjerid, S., Lin, T.: A p-th degree immersed finite element for boundary value problems with discontinuous coefficients. Appl. Numer Math. 59(6), 1303–1321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adjerid, S., Massey, T.C.: Superconvergence Of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput. Methods Appl. Mech. Engrg. 195, 3331–3346 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Strouboulis, T.: The finite element method and its reliability. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York (2001)

    MATH  Google Scholar 

  5. Babuška, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K.: Computer-based proof of the existence of superconvergence points in the finite element method: superconvergence of the derivatives in finite element solutions of Laplace’s, Poisson’s, and the elasticity equations. Numer. Meth. PDEs. 12, 347–392 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramble, J., Schatz, A.: High order local accuracy by averaging in the finite element method. Math. Comp. 31, 94–111 (1997)

    Article  MATH  Google Scholar 

  7. Brenner, S.C., Ridgway, S.L.: The Mathematical Theory of Finite Element Methods Texts in Applied Mathematics, vol. 15. Springer-Verlag, New York (1994)

  8. Cai, Z.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Camp, B., Lin, T., Lin, Y., Sun, W.: Quadratic immersed finite element spaces and their approximation capabilities. Adv. Comput. Math. 24(1-4), 81–112 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, W., Shu, C.-W., Yang, Y., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for 2-D hyperbolic equations. SIAM J. Numer. Anal. 53, 1651–1671 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cao, W., Zhang, Z.: Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations. Math. Comp. 85, 63–84 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cao, W., Zhang, Z., Zou, Q.: Superconvergence of any order finite volume schemes for 1D general elliptic equations. J. Sci. Comput. 56, 566–590 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cao, W., Zhang, Z., Zou, Q.: Superconvergence of Discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 52, 2555–2573 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cao, W., Zhang, Z., Zou, Q.: Is 2k-conjecture valid for finite volume methods?. SIAM J. Numer. Anal. 53(2), 942–962 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Chou, S., Ye, X.: Superconvergence Of finite volume methods for the second order elliptic problem. Comput. Methods Appl. Mech. Eng. 196, 3706–3712 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chou, S.-H., Kwak, D.Y., Wee, K.T.: Optimal convergence analysis of an immersed interface finite element method. Adv. Comput. Math. 33(2), 149–168 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Douglas, J.Jr., 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 

  19. Gong, Y., Li, B., Li, Z.: Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions. SIAM J. Numer. Anal. 46(1), 472–495 (2007/08)

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

    Article  MathSciNet  MATH  Google Scholar 

  21. He, X., Lin, T., Lin, Y.: Approximation capability of a bilinear immersed finite element space. Numer. Methods Partial Differential Equations 24(5), 1265–1300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, X., Lin, T., Lin, Y., Zhang, X: Immersed finite element methods for parabolic equations with moving interface. Numer. Methods Partial Differential Equations 29(2), 619–646 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kafafy, R., Lin, T., Lin, Y., Wang, J: Three-dimensional immersed finite element methods for electric field simulation in composite materials. Internat. J. Numer. Methods Engrg. 64(7), 940–972 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kr̆iz̆ek, M., Neittaanmäki, P.: On superconvergence techniques. Acta Appl. Math. 9, 175–198 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, Z.: The immersed interface method using a finite element formulation. Appl. Numer. Math. 27(3), 253–267 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approximation capability. Numer. Methods Partial Differential Equations 20 (3), 338–367 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96(1), 61–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lin, T., Lin, Y., Zhang, X.: Immersed finite element method of lines for moving interface problems with nonhomogeneous flux jump. In: Recent advances in scientific computing and applications, volume 586 of Contemp. Math., pp 257–265, Providence, RI (2013)

  29. Lin, T., Lin, Y., X. Zhang: A method of lines based on immersed finite elements for parabolic moving interface problems. Adv. Appl. Math Mech. 5(4), 548–568 (2013)

    Article  MathSciNet  Google Scholar 

  30. Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer Anal. 53(2), 1121–1144 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lin, T., Sheen, D., Zhang, X.: A locking-free immersed finite element method for planar elasticity interface problems. J. Comput. Phys. 247, 228–247 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lin, T., Yang, Q., Zhang, X.: A Priori error estimates for some discontinuous Galerkin, immersed finite element methods. J. Sci. Comput. 65(3), 875–894 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lin, T., Yang, Q., Zhang, X.: Partially penalized immersed finite element methods for parabolic interface problems. Numer. Methods Partial Differential Equations 31(6), 1925–1947 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Schatz, A.H., Sloan, I.H., Wahlbin, L.B.: Superconvergence in finite element methods and meshes which are symmetric with respect to a point. SIAM J. Numer. Anal. 33, 505–521 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shen, J., Tang, T., Wang, L.-L.: Spectral methods, volume 41 of Springer Series in Computational Mathematics. Springer, Heidelberg (2011). Algorithms, analysis and applications

    Google Scholar 

  36. Thomee, V.: High order local approximation to derivatives in the finite element method. Math. Comp. 31, 652–660 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Vallaghé, S., Papadopoulo, T.: A trilinear immersed finite element method for solving the electroencephalography forward problem. SIAM J. Sci. Comput. 32 (4), 2379–2394 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wahlbin, L.B.: Superconvergence in Galerkin Finite Element Methods, volume 1605 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1995)

    Google Scholar 

  39. Xie, Z., Zhang, Z.: Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D. Math. Comp. 79, 35–45 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xu, J., Zou, Q.: Analysis of linear and quadratic simplitical finite volume methods for elliptic equations. Numer. Math. 111, 469–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110–3133 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, Z.: Superconvergence of Spectral collocation and p-version methods in one dimensional problems. Math. Comp. 74, 1621–1636 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, Z.: Superconvergence of a Chebyshev spectral collocation method. J. Sci. Comput. 34, 237–246 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, Z.: Superconvergence points of polynomial spectral interpolation. SIAM J. Numer. Anal. 50, 2966–2985 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zienkiewicz, O.C., Cheung, Y.K.: The Finite Element Method in Structural and Continuum Mechanics: Numerical Solution of Problems in Structural and Continuum Mechanics. Vol 1, European Civil Engineering Series, McGraw-Hill (1967)

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Correspondence to Xu Zhang.

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Communicated by: Jan Hesthaven

This work is supported in part by the China Postdoctoral Science Foundation 2015M570026, and the National Natural Science Foundation of China (NSFC) under grants No. 91430216, 11471031, 11501026, and the US National Science Foundation (NSF) through grant DMS-1419040.

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Cao, W., Zhang, X. & Zhang, Z. Superconvergence of immersed finite element methods for interface problems. Adv Comput Math 43, 795–821 (2017). https://doi.org/10.1007/s10444-016-9507-7

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  • DOI: https://doi.org/10.1007/s10444-016-9507-7

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