Abstract
In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and DG formulation in these methods allows local mesh refinement in the Cartesian mesh structure for interface problems. Numerical results are provided to demonstrate the convergence and local mesh refinement features of these DG-IFE methods.
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Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)
Arnold, D.N., Brezzi, F., Cockburn, B.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)
Babuška, I., Osborn, J.E.: Can a finite element method perform arbitrarily badly? Math. Comp. 69(230), 443–462 (2000)
Babuška, I., Zlámal, M.: Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10, 863–875 (1973)
Birdsall, C.K., Langdon, A.B.: Plasma Physics via Computer Simulation (Series in Plasma Physics). Institute of Physisc Publishing, London (1991)
Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6(2), 109–138 (1996)
Cai, Z., Ye, X., Zhang, S.: Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations. SIAM J. Numer. Anal. 49(5), 1761–1787 (2011)
Cai, Z., Zhang, S.: Flux recovery and a posteriori error estimators: conforming elements for scalar elliptic equations. SIAM J. Numer. Anal. 48(2), 578–602 (2010)
Chen, Z.: Finite element methods and their applications. Scientific Computation. Springer, Berlin (2005)
Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998)
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)
Cockburn, B., Karniadakis, G.E., Shu, C.-W., (eds): Discontinuous Galerkin methods, volume 11 of Lecture Notes in Computational Science and Engineering. Springer, Berlin, 2000. Theory, computation and applications, Papers from the 1st International Symposium held in Newport, RI, May 24–26 (1999)
Douglas, J., Jr., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. In Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975), pages 207–216. Lecture Notes in Phys., Vol. 58. Springer, Berlin (1976)
Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152(2), 457–492 (1999)
Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002)
He, X.: Bilinear immersed finite elements for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2009)
He, X., Lin, T., Lin, Y.: Approximation capability of a bilinear immersed finite element space. Numer. Methods Partial Differ. Equ. 24(5), 1265–1300 (2008)
He, X., Lin, T., Lin, Y.: Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient. J. Syst. Sci. Complex. 23(3), 467–483 (2010)
He, X., Lin, T., Lin, Y.: The convergence of the bilinear and linear immersed finite element solutions to interface problems. Numer. Methods Partial Differ. Equ. 28(1), 312–330 (2012)
He, X., Lin, T., Lin, Y.: A selective immersed discontinuous Galerkin method for elliptic interface problems. Math. Methods Appl. Sci. 37(7), 983–1002 (2014)
He, X., Lin, T., Lin, Y., Zhang, X.: Immersed finite element methods for parabolic equations with moving interface. Numer. Methods Partial Differ. Equ. 29(2), 619–646 (2013)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods, Volume 54 of Texts in Applied Mathematics. Springer, New York (2008) Algorithms, analysis, and applications
Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)
Kafafy, R., Wang, J.: Whole ion optics gridlet simulations using a hybrid-grid immersed-finite-element particle-in-cell code. J. Propuls. Power 23(1), 59–68 (2007)
Kwak, D.Y., Wee, K.T., Chang, K.S.: An analysis of a broken \(P_1\)-nonconforming finite element method for interface problems. SIAM J. Numer. Anal. 48(6), 2117–2134 (2010)
Li, Z.: The immersed interface method using a finite element formulation. Appl. Numer. Math. 27(3), 253–267 (1998)
Li, Z., Ito, K.: The immersed interface method, volume 33 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2006). Numerical solutions of PDEs involving interfaces and irregular domains
Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approximation capability. Numer. Methods Partial Differ. Equ. 20(3), 338–367 (2004)
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)
Lin, T., Lin, Y., Rogers, R., Ryan, M.L.: A rectangular immersed finite element space for interface problems. In Scientific computing and applications (Kananaskis, AB, 2000), volume 7 of Adv. Comput. Theory Pract., pages 107–114. Nova Sci. Publ., Huntington, NY (2001)
Lin, T., Lin, Y., Zhang, X.: A method of lines based on immersed finite elements for parabolic moving interface problems. Adv. Appl. Math. Mech. 5(4), 548–568 (2013)
Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. (2014) (accepted)
Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999)
Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)
Rangarajan, R., Lew, A.J.: Universal meshes: A method for triangulating planar curved domains immersed in nonconforming meshes. Int. J. Numer. Methods Eng. 98(4), 236–264 (2014)
Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM (1973)
Rivière, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations, volume 35 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008). Theory and implementation
Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci., 3(3–4), 337–360 (1999)
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)
Wang, J., He, X., Cao, Y.: Modeling electrostatic levitation of dust particles on lunar surface. IEEE Trans. Plasma Sci. 36(5), 2459–2466 (2008)
Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)
Zhang, X.: Nonconforming Immersed Finite Element Methods for Interface Problems. ProQuest LLC, Ann Arbor, MI, (2013). Thesis (Ph.D.)-Virginia Polytechnic Institute and State University
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The authors would like to thank anonymous referees whose comments and suggestions enhanced the presentation of our research work.
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This work is partially supported by NSF grant DMS-1016313.
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Lin, T., Yang, Q. & Zhang, X. A Priori Error Estimates for Some Discontinuous Galerkin Immersed Finite Element Methods. J Sci Comput 65, 875–894 (2015). https://doi.org/10.1007/s10915-015-9989-3
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DOI: https://doi.org/10.1007/s10915-015-9989-3