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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
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.
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.
C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 1986, 46(173): 1–26.
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.
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.
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.
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.
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.
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.
E. Burman and B. Stamm, Local discontinuous Galerkin method with reduced stabilization for diffusion equations, Commun. Comput. Phys., 2009, 5(2–4): 498–514.
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.
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.
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.
J. Haink and C. Rohde, Local discontinuous-Galerkin schemes for model problems in phase transition theory, Commun. Comput. Phys., 2008, 4(4): 860–893.
G. Kanschat, Block preconditioners for LDG discretizations of linear incompressible flow problems, J. Sci. Comput., 2005, 22/23: 371–384.
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.
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.
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.
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.
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.
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.
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.
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.
J. Peraire and P. O. Persson, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 2008, 30(4): 1806–1824.
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.
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.
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.
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.
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.
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.
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.
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.
D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal, 1982, 19:742–760.
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.
J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lecture Notes in Physics, 1976, 58: 207–216.
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.
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.
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.
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.
M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal, 1978, 15: 152–161.
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.
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.
Z. Chen, Finite Element Methods and Their Applications, Scientific Computation, Springer-Verlag, Berlin, 2005.
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.
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.
I. Babuska, The finite element method with penalty, Math. Comp., 1973, 27: 221–228.
I. Babuska and M. Zlamal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal., 1973, 10: 863–875.
M. Delves and C. A. Hall, An implicit matching principle for global element calculations, J. Inst. Math. Appl, 1979, 23: 223–234.
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.
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.
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.
B. Heinrich, Finite Difference Methods on Irregular Networks, volume 82 of International Series of Numerical Mathematics, Birkhäuser, Boston, 1987.
I. Babuška, The finite element method for elliptic equations with discontinuous coefficients, Computing, 1970, 5: 207–213.
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.
Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 1998, 79: 175–202.
I. Babuška and J. E. Osborn, Can a finite element method perform arbitrarily badly? Math. Comp., 2000, 69(230): 443–462.
D. W. Hewitt, The embedded curved boundary method for orthogonal simulation meshes, J. Comput. Phys., 1997, 138: 585–616.
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.
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.
C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 1977, 25: 220–252.
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.
I. Babuška and J. Melenk, The partition of unity method, Int. J. Numer. Meth. Eng., 1997, 40: 727–758.
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.
T. Belytschko, N. Moës, S. Usui, and C. Primi, Arbitrary discontinuities in finite elements, Int. J. Numer. Meth. Eng., 2001, 50: 993–1013.
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.
S. Adjerid and T. Lin, Higher-order immersed discontinuous Galerkin methods, Int. J. Inf. Syst. Sci., 2007, 3(4): 555–568.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Z. Li, The immersed interface method using a finite element formulation, Appl. Numer. Math., 1997, 27(3): 253–267.
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.
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.
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.
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.
S. A. Sauter and R. Warnke, Composite finite elements for elliptic boundary value problems with discontinuous coefficients, Computing, 2006, 77(1): 29–55.
T. S. Wang, A Hermite cubic immersed finite element space for beam design problems, Master’s thesis, Virginia Polytechnic Institute and State University, 2005.
X. M. He, Bilinear immersed finite elements for interface problems, Ph.D. dissertation, Virginia Polytechnic Institute and State University, 2009.
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.
This research is supported by NSF grant DMS-0713763, the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 501709), the AMSS-PolyU Joint Research Institute for Engineering and Management Mathematics, and NSERC (Canada), and this article is dedicated to Professor David Russell’s 70th birthday.
Rights and permissions
About this article
Cite this article
He, X., Lin, T. & Lin, Y. Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient. J Syst Sci Complex 23, 467–483 (2010). https://doi.org/10.1007/s11424-010-0141-z
- Adaptive mesh
- discontinuous Galerkin
- immersed interface
- interface problems