Advertisement

Journal of Systems Science and Complexity

, Volume 23, Issue 3, pp 467–483 | Cite as

Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient

  • Xiaoming HeEmail author
  • Tao Lin
  • Yanping Lin
Article

Abstract

This paper applies bilinear immersed finite elements (IFEs) in the interior penalty discontinuous Galerkin (DG) methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bilinear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.

Key words

Adaptive mesh discontinuous Galerkin immersed interface interface problems penalty 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report No. LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
  2. [2]
    P. Lasaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in Partial Differential Equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Academic Press, New York, 1974, 89–123.Google Scholar
  3. [3]
    C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 1986, 46(173): 1–26.zbMATHMathSciNetGoogle Scholar
  4. [4]
    B. Cockburn and J. Guzmán, Error estimates for the Runge-Kutta discontinuous Galerkin method for the transport equation with discontinuous initial data, SIAM J. Numer. Anal., 2008, 46(3): 1364–1398.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    B. Cockburn, S. C. Hou, and C. W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, IV. The multidimensional case, Math. Comp., 1990, 54(190): 545–581.zbMATHMathSciNetGoogle Scholar
  6. [6]
    B. Cockburn and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, II. General framework, Math. Comp., 1989, 52(186): 411–435.zbMATHMathSciNetGoogle Scholar
  7. [7]
    J. Qiu, T. Liu, and B. C. Khoo, Simulations of compressible two-medium flow by Runge-Kutta discontinuous Galerkin methods with the ghost fluid method, Commun. Comput. Phys., 2008, 3(2): 479–504.zbMATHMathSciNetGoogle Scholar
  8. [8]
    H. Zhu and J. Qiu, Adaptive Runge-Kutta discontinuous Galerkin methods using different indicators: one-dimensional case, J. Comput. Phys., 2009, 228(18): 6957–6976.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Zhu, J. Qiu, C. W. Shu, and M. Dumbser, Runge-Kutta discontinuous Galerkin method using WENO limiters, II. Unstructured meshes, J. Comput. Phys., 2008, 227(9): 4330–4353.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    E. Burman and B. Stamm, Local discontinuous Galerkin method with reduced stabilization for diffusion equations, Commun. Comput. Phys., 2009, 5(2–4): 498–514.MathSciNetGoogle Scholar
  11. [11]
    B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 1998, 35(6): 2440–2463.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    B. Dong and C. W. Shu, Analysis of a local discontinuous Galerkin method for linear timedependent fourth-order problems, SIAM J. Numer. Anal., 2009, 47(5): 3240–3268.CrossRefMathSciNetGoogle Scholar
  13. [13]
    J. Guzmán, Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem, Math. Comp., 2008, 77(263): 1293–1322.CrossRefMathSciNetGoogle Scholar
  14. [14]
    J. Haink and C. Rohde, Local discontinuous-Galerkin schemes for model problems in phase transition theory, Commun. Comput. Phys., 2008, 4(4): 860–893.MathSciNetGoogle Scholar
  15. [15]
    G. Kanschat, Block preconditioners for LDG discretizations of linear incompressible flow problems, J. Sci. Comput., 2005, 22/23: 371–384.CrossRefMathSciNetGoogle Scholar
  16. [16]
    W. Wang and C. W. Shu, The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode, J. Sci. Comput., 2009, 40(1–3): 360–374.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Y. Xu and C. W. Shu, Local discontinuous Galerkin method for the Hunter-Saxton equation and its zero-viscosity and zero-dispersion limits, SIAM J. Sci. Comput., 2008, 31(2): 1249–1268.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    K. Fan, W. Cai, and X. Ji, A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides, J. Comput. Phys., 2008, 227(15): 7178–7191.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    K. Fan, W. Cai, and X. Ji, A generalized discontinuous Galerkin (GDG) method for schrödinger equations with nonsmooth solutions, J. Comput. Phys., 2008, 227(4): 2387–2410.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    V. Girault, S. Sun, M. F. Wheeler, and I. Yotov, Coupling discontinuous Galerkin and mixed finite element discretizations using mortar finite elements, SIAM J. Numer. Anal., 2008, 46(2): 949–979.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    S. Kumar, N. Nataraj, and A. K. Pani, Discontinuous galerkin finite volume element methods for second-order linear elliptic problems, Numer. Methods Partial Differential Equations, 2009, 25(6): 1402–1424.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    K. Wang, A uniform optimal-order estimate for an Eulerian-Lagrangian discontinuous Galerkin method for transient advection-diffusion equations, Numer. Methods Partial Differential Equations, 2009, 25(1): 87–109.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    K. Wang, H. Wang, M. Al-Lawatia, and H. Rui, A family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations and their optimal-order L 2 error estimates, Commun. Comput. Phys., 2009, 6(1): 203–230.CrossRefMathSciNetGoogle Scholar
  24. [24]
    J. Peraire and P. O. Persson, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 2008, 30(4): 1806–1824.CrossRefMathSciNetGoogle Scholar
  25. [25]
    P. E. Bernard, J. F. Remacle, R. Comblen, V. Legat, and K. Hillewaert, High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations, J. Comput. Phys., 2009, 228(17): 6514–6535.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    J. K. Djoko, Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds, Numer. Methods Partial Differential Equations, 2008, 24(1): 296–311.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    L. Pesch and J. J. W. van der Vegt, A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids, J. Comput. Phys., 2008, 227(11): 5426–5446.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    F. Prill, M. Lukáčová-Medvidová, and R. Hartmann, Smoothed aggregation multigrid for the discontinuous Galerkin method, SIAM J. Sci. Comput., 2009, 31(5): 3503–3528.CrossRefMathSciNetGoogle Scholar
  29. [29]
    M. Restelli and F. X. Giraldo, A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling, SIAM J. Sci. Comput., 2009, 31(3): 2231–2257.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    G. Sun, D. Liang, and W. Wang, Numerical analysis to discontinuous Galerkin methods for the age structured population model of marine invertebrates, Numer. Methods Partial Differential Equations, 2009, 25(2): 470–493.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    Z. Xu, Y. Liu, and C. W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, J. Comput. Phys., 2009, 228(6): 2194–2212.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    B. Cockburn, G. E. Karniadakis, and C. W. Shu, The development of discontinuous Galerkin methods. Discontinuous Galerkin methods (Newport, RI, 1999), Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2000, 11: 3–50.MathSciNetGoogle Scholar
  33. [33]
    D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal, 1982, 19:742–760.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    I. Babuska, C. E. Baumann, and J. T. Oden, A discontinuous hp finite element method for diffusion problems: 1D analysis, Comput. & Math. Appl., 1999, 37: 103–122.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lecture Notes in Physics, 1976, 58: 207–216.CrossRefMathSciNetGoogle Scholar
  36. [36]
    J. T. Oden, I. Babuska, and C. E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys, 1998, 146: 491–519.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    B. Rivière, M. F. Wheeler, and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems Part 1, Comput. Geosci., 1999, 3: 337–360.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    T. Sun, Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem, Int. J. Numer. Anal. Mod., 2010, 7(1): 87–107.Google Scholar
  39. [39]
    T. Sun and D. Yang, Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations, Numer. Methods Partial Differential Equations, 2008, 24(3): 879–896.zbMATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal, 1978, 15: 152–161.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    H. Chen and Z. Chen, Stability and convergence of mixed discontinuous finite element methods for second-order differential problems, J. Numer. Math., 2003, 11(4): 253–287.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    H. Chen, Z. Chen, and B. Li, Numerical study of the hp version of mixed discontinuous finite element methods for reaction-diffusion problems: the 1D case, Numer. Methods Partial Differential Equations, 2003, 19(4): 525–553.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    Z. Chen, Finite Element Methods and Their Applications, Scientific Computation, Springer-Verlag, Berlin, 2005.Google Scholar
  44. [44]
    D. N. Arnold, F. Brezzi, B. Cockburn L. D. and Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 2001, 39(5): 1749–1779.CrossRefMathSciNetGoogle Scholar
  45. [45]
    Z. Chen, On the relationship of various discontinuous finite element methods for second-order elliptic equations, East-West J. Numer. Math., 2001, 9(2): 99–122.zbMATHMathSciNetGoogle Scholar
  46. [46]
    I. Babuska, The finite element method with penalty, Math. Comp., 1973, 27: 221–228.zbMATHMathSciNetGoogle Scholar
  47. [47]
    I. Babuska and M. Zlamal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal., 1973, 10: 863–875.zbMATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    M. Delves and C. A. Hall, An implicit matching principle for global element calculations, J. Inst. Math. Appl, 1979, 23: 223–234.zbMATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    M. J. Grote, A. Schneebeli, and D. Schötzau, Interior penalty discontinuous Galerkin method for Maxwell’s equations: energy norm error estimates, J. Comput. Appl. Math., 2007, 204(2): 375–386.zbMATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    T. J. R. Hughes, G. Engel, L. Mazzei, and M. G. Larson, A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency, in Discontinuous Galerkin Methods, Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, 2000, 11: 135–146.MathSciNetGoogle Scholar
  51. [51]
    G. Kanschat and R. Rannacher, Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems, J. Numer. Math., 2002, 10(4): 249–274.zbMATHMathSciNetGoogle Scholar
  52. [52]
    B. Heinrich, Finite Difference Methods on Irregular Networks, volume 82 of International Series of Numerical Mathematics, Birkhäuser, Boston, 1987.Google Scholar
  53. [53]
    I. Babuška, The finite element method for elliptic equations with discontinuous coefficients, Computing, 1970, 5: 207–213.zbMATHCrossRefGoogle Scholar
  54. [54]
    J. H. Bramble and J. T. King, A finite element method for interface problems in domains with smooth boundary and interfaces, Adv. Comput. Math., 1996, 6: 109–138.CrossRefMathSciNetGoogle Scholar
  55. [55]
    Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 1998, 79: 175–202.zbMATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    I. Babuška and J. E. Osborn, Can a finite element method perform arbitrarily badly? Math. Comp., 2000, 69(230): 443–462.zbMATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    D. W. Hewitt, The embedded curved boundary method for orthogonal simulation meshes, J. Comput. Phys., 1997, 138: 585–616.CrossRefMathSciNetGoogle Scholar
  58. [58]
    D. M. Ingram, D. M. Causon, and C. G. Mingham, Developments in Cartesian cut cell methods, Math. Comput. Simulation, 2003, 61(3–6): 561–572.zbMATHCrossRefMathSciNetGoogle Scholar
  59. [59]
    Z. Li and K. Ito, The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, Vol 33 of Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 2006.Google Scholar
  60. [60]
    C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 1977, 25: 220–252.zbMATHCrossRefMathSciNetGoogle Scholar
  61. [61]
    Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys, 2006, 213(1): 1–30.zbMATHCrossRefMathSciNetGoogle Scholar
  62. [62]
    I. Babuška and J. Melenk, The partition of unity method, Int. J. Numer. Meth. Eng., 1997, 40: 727–758.zbMATHCrossRefGoogle Scholar
  63. [63]
    I. Babuška and J. E. Osborn. Finite element methods for the solution of problems with rough input data. In P. Grisvard, W. Wendland, and J.R. Whiteman, editors, Singular and Constructive Methods for their Treatment, Lecture Notes in Mathematics, New York, Springer-Verlag, 1985, 1–18.CrossRefGoogle Scholar
  64. [64]
    T. Belytschko, N. Moës, S. Usui, and C. Primi, Arbitrary discontinuities in finite elements, Int. J. Numer. Meth. Eng., 2001, 50: 993–1013.zbMATHCrossRefGoogle Scholar
  65. [65]
    R. E. Ewing, H. Wang, and T.F. Russell, Eulerian-Lagrangian localized adjoint methods fo convection-diffusion equations and their convergence analysis, IMA J. Numer. Anal., 1995, 15: 405–495.zbMATHCrossRefMathSciNetGoogle Scholar
  66. [66]
    S. Adjerid and T. Lin, Higher-order immersed discontinuous Galerkin methods, Int. J. Inf. Syst. Sci., 2007, 3(4): 555–568.zbMATHMathSciNetGoogle Scholar
  67. [67]
    S. Adjerid and T. Lin, p-th degree immersed finite element for boundary value problems with discontinuous coefficients, Appl. Numer. Math, 2009, 59(6): 1303–1321.zbMATHCrossRefMathSciNetGoogle Scholar
  68. [68]
    B. Camp, T. Lin, Y. Lin, and W. Sun, Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math., 2006, 24(1–4): 81–112.zbMATHCrossRefMathSciNetGoogle Scholar
  69. [69]
    R. E. Ewing, Z. Li, T. Lin, and Y. Lin, The immersed finite volume element methods for the elliptic interface problems, Modelling’98 (prague), Math. Comput. Simulation, 1999, 50(1–4): 63–76.zbMATHCrossRefMathSciNetGoogle Scholar
  70. [70]
    Y. Gong, B. Li, and Z. Li, Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions, SIAM J. Numer. Anal., 2008, 46: 472–495.zbMATHCrossRefMathSciNetGoogle Scholar
  71. [71]
    X. M. He, T. Lin, and Y. Lin, Approximation capability of a bilinear immersed finite element space, Numer. Methods Partial Differential Equations, 2008, 24(5): 1265–1300.zbMATHCrossRefMathSciNetGoogle Scholar
  72. [72]
    X. M. He, T. Lin, and Y. Lin, A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficients, Commun. Comput. Phys., 2009, 6(1): 185–202.CrossRefMathSciNetGoogle Scholar
  73. [73]
    X. M. He, T. Lin, and Y. Lin, Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions, Int. J. Numer. Anal. Model., accepted, 2010.Google Scholar
  74. [74]
    R. Kafafy, T. Lin, Y. Lin, and J. Wang, Three-dimensional immersed finite element methods for electric field simulation in composite materials, Int. J. Numer. Meth. Engrg., 2005, 64(7): 940–972.zbMATHCrossRefMathSciNetGoogle Scholar
  75. [75]
    R. Kafafy, J. Wang, and T. Lin, A hybrid-grid immersed-finite-element particle-in-cell simulation model of ion optics plasma dynamics, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 2005, 12: 1–16.MathSciNetGoogle Scholar
  76. [76]
    Z. Li, The immersed interface method using a finite element formulation, Appl. Numer. Math., 1997, 27(3): 253–267.CrossRefGoogle Scholar
  77. [77]
    Z. Li, T. Lin, Y. Lin, and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods Partial Differential Equations, 2004, 20(3): 338–367.zbMATHCrossRefMathSciNetGoogle Scholar
  78. [78]
    Z. Li, T. Lin, and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 2003, 96(1): 61–98.zbMATHCrossRefMathSciNetGoogle Scholar
  79. [79]
    T. Lin, Y. Lin, R. C. Rogers, and L. M. Ryan, A rectangular immersed finite element method for interface problems, in P. Minev and Y. Lin, editors, Advances in Computation: Theory and Practice, Nova Science Publishers, Inc., 2001, 7: 107–114.Google Scholar
  80. [80]
    T. Lin, Y. Lin, and W. Sun, Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems, Discrete Contin. Dyn. Syst. Ser. B, 2007, 7(4): 807–823.zbMATHMathSciNetGoogle Scholar
  81. [81]
    S. A. Sauter and R. Warnke, Composite finite elements for elliptic boundary value problems with discontinuous coefficients, Computing, 2006, 77(1): 29–55.zbMATHCrossRefMathSciNetGoogle Scholar
  82. [82]
    T. S. Wang, A Hermite cubic immersed finite element space for beam design problems, Master’s thesis, Virginia Polytechnic Institute and State University, 2005.Google Scholar
  83. [83]
    X. M. He, Bilinear immersed finite elements for interface problems, Ph.D. dissertation, Virginia Polytechnic Institute and State University, 2009.Google Scholar
  84. [84]
    T. Lin, Y. Lin, R. C. Rogers, and L. M. Ryan, A rectangular immersed finite element method for interface problems, in P. Minev and Y. Lin, Editors, Advances in Computation: Theory and Practice, Nova Science Publishers, Inc., 2001, 7: 107–114.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Department of Applied MathematicsHong Kong Polytechnic UniversityHung Hom, Hong KongChina
  3. 3.Department of Mathematical and Statistics ScienceUniversity of AlbertaEdmontonCanada

Personalised recommendations