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A discontinuous Galerkin method by patch reconstruction for elliptic interface problem on unfitted mesh. (English) Zbl 1440.65218

Summary: We propose a discontinuous Galerkin (DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both the \(L^2\) norm and the DG energy norm are obtained, without restrictions on how the interface intersects the elements in the mesh. The stability near the interface is ensured by the patch reconstruction and no special numerical flux is required. The convergence order by numerical results in both two and three dimensions agrees with the error estimates perfectly. More than enjoying the advantages of the DG method, the new method may achieve even better efficiency in number of degrees of freedom than the conforming finite element method as illustrated by our numerical examples.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Software:

IIMPACK; PolyMesher
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Full Text: DOI arXiv

References:

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amsterdam) 140, Elsevier/Academic Press, Amsterdam, 2003. · Zbl 1098.46001
[2] S. Adjerid, N. Chaabane, and T. Lin, An immersed discontinuous finite element method for Stokes interface problems, Comput. Methods Appl. Mech. Engrg., 293 (2015), pp. 170-190, https://doi.org/10.1016/j.cma.2015.04.006. · Zbl 1423.76200
[3] N. An and H.-z. Chen, A partially penalty immersed interface finite element method for anisotropic elliptic interface problems, Numer. Methods Partial Differential Equations, 30 (2014), pp. 1984-2028, https://doi.org/10.1002/num.21886. · Zbl 1312.65179
[4] P. F. Antonietti, L. Beira͂o da Veiga, and M. Verani, A mimetic discretization of elliptic obstacle problems, Math. Comp., 82 (2013), pp. 1379-1400, https://doi.org/10.1090/S0025-5718-2013-02670-1. · Zbl 1271.65136
[5] I. Babuka, The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), pp. 207-213.
[6] J. W. Barrett and C. M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal., 7 (1987), pp. 283-300, https://doi.org/10.1093/imanum/7.3.283. · Zbl 0629.65118
[7] T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Internat. J. Numer. Methods Engrg., 45 (1999), pp. 601-620. · Zbl 0943.74061
[8] E. Burman and A. Ern, An unfitted hybrid high-order method for elliptic interface problems, SIAM J. Numer. Anal., 56 (2018), pp. 1525-1546, https://doi.org/10.1137/17M1154266. · Zbl 1448.65201
[9] A. Cangiani, E. H. Georgoulis, and Y. A. Sabawi, Adaptive discontinuous Galerkin methods for elliptic interface problems, Math. Comp., 87 (2018), pp. 2675-2707, https://doi.org/10.1090/mcom/3322. · Zbl 1397.65252
[10] W. Cao, X. Zhang, Z. Zhang, and Q. Zou, Superconvergence of immersed finite volume methods for one-dimensional interface problems, J. Sci. Comput., 73 (2017), pp. 543-565, https://doi.org/10.1007/s10915-017-0532-6. · Zbl 1384.65045
[11] T. Chen and J. Strain, Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems, J. Comput. Phys., 227 (2008), pp. 7503-7542, https://doi.org/10.1016/j.jcp.2008.04.027. · Zbl 1157.65064
[12] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), pp. 175-202, https://doi.org/10.1007/s002110050336. · Zbl 0909.65085
[13] C.-C. Chu, I. G. Graham, and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), pp. 1915-1955, https://doi.org/10.1090/S0025-5718-2010-02372-5. · Zbl 1202.65154
[14] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978, http://dx.doi.org/10.1115/1.3424474. · Zbl 0383.65058
[15] T. Cui, W. Leng, H. Liu, L. Zhang, and W. Zheng, High-order numerical quadratures in a tetrahedron with an implicitly defined curved interface, ACM Trans. Math. Software, 46 (2020), https://lsec.cc.ac.cn/phg/download.htm. · Zbl 1484.65046
[16] R. P. Fedkiw, T. Aslam, B. Merriman, and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), pp. 457-492, https://doi.org/10.1006/jcph.1999.6236. · Zbl 0957.76052
[17] R. Guo and T. Lin, A higher degree immersed finite element method based on a Cauchy extension for elliptic interface problems, SIAM J. Numer. Anal., 57 (2019), pp. 1545-1573, https://doi.org/10.1137/18M121318X. · Zbl 1420.65122
[18] G. Guyomarc’h, C.-O. Lee, and K. Jeon, A discontinuous Galerkin method for elliptic interface problems with application to electroporation, Comm. Numer. Methods Engrg., 25 (2009), pp. 991-1008, https://doi.org/10.1002/cnm.1132. · Zbl 1175.65136
[19] J. Guzmán, M. A. Sánchez, and M. Sarkis, A finite element method for high-contrast interface problems with error estimates independent of contrast, J. Sci. Comput., 73 (2017), pp. 330-365, https://doi.org/10.1007/s10915-017-0415-x. · Zbl 1380.65369
[20] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5537-5552, https://doi.org/10.1016/S0045-7825(02)00524-8. · Zbl 1035.65125
[21] S. Hou and X.-D. Liu, A numerical method for solving variable coefficient elliptic equation with interfaces, J. Comput. Phys., 202 (2005), pp. 411-445, https://doi.org/10.1016/j.jcp.2004.07.016. · Zbl 1061.65123
[22] S. Hou, W. Wang, and L. Wang, Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces, J. Comput. Phys., 229 (2010), pp. 7162-7179, https://doi.org/10.1016/j.jcp.2010.06.005. · Zbl 1197.65183
[23] P. Huang, H. Wu, and Y. Xiao, An unfitted interface penalty finite element method for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 323 (2017), pp. 439-460, https://doi.org/10.1016/j.cma.2017.06.004. · Zbl 1439.74422
[24] 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 (Newport, RI, 1999), Lect. Notes Comput. Sci. Eng. 11, Springer, Berlin, 2000, pp. 135-146, https://doi.org/10.1007/978-3-642-59721-3_9. · Zbl 0946.65109
[25] L. N. T. Huynh, N. C. Nguyen, J. Peraire, and B. C. Khoo, A high-order hybridizable discontinuous Galerkin method for elliptic interface problems, Internat. J. Numer. Methods Engrg., 93 (2013), pp. 183-200, https://doi.org/10.1002/nme.4382. · Zbl 1352.65513
[26] R. B. Kellogg, Higher order singularities for interface problems, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, 1972, pp. 589-602, https://doi.org/10.1007/bf01932971.
[27] R. B. Kellogg, On the Poisson equation with intersecting interfaces, Appl. Anal., 4 (1974), pp. 101-129, https://doi.org/10.1080/00036817408839086. · Zbl 0307.35038
[28] R. J. LeVeque and Z. L. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 1019-1044, https://doi.org/10.1137/0731054. · Zbl 0811.65083
[29] R. Li, P. Ming, Z. Sun, F. Yang, and Z. Yang, A discontinuous Galerkin method by patch reconstruction for biharmonic problem, J. Comput. Math., 37 (2019), pp. 563-580. · Zbl 1449.65321
[30] R. Li, P. Ming, Z. Sun, and Z. Yang, An arbitrary-order discontinuous Galerkin method with one unknown per element, J. Sci. Comput., 80 (2019), pp. 268-288, https://doi.org/10.1007/s10915-019-00937-y. · Zbl 1416.74088
[31] R. Li, P. Ming, and F. Tang, An efficient high order heterogeneous multiscale method for elliptic problems, Multiscale Model. Simul., 10 (2012), pp. 259-283, https://doi.org/10.1137/110836626. · Zbl 1246.65209
[32] Z. Li, The immersed interface method using a finite element formulation, Appl. Numer. Math., 27 (1998), pp. 253-267, https://doi.org/10.1016/S0168-9274(98)00015-4. · Zbl 0936.65091
[33] Z. Li and K. Ito, The Immersed Interface method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains , Frontiers in Appl. Math. 33, SIAM, Philadelphia, 2006, https://doi.org/10.1137/1.9780898717464. · Zbl 1122.65096
[34] T. Lin, Y. Lin, and X. Zhang, Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal., 53 (2015), pp. 1121-1144, https://doi.org/10.1137/130912700. · Zbl 1316.65104
[35] X.-D. Liu, R. P. Fedkiw, and M. Kang, A boundary condition capturing method for Poisson’s equation on irregular domains, J. Comput. Phys., 160 (2000), pp. 151-178, https://doi.org/10.1006/jcph.2000.6444. · Zbl 0958.65105
[36] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes, A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61 (2014), pp. 604-628, https://doi.org/10.1007/s10915-014-9838-9. · Zbl 1417.76028
[37] R. Massjung, An unfitted discontinuous Galerkin method applied to elliptic interface problems, SIAM J. Numer. Anal., 50 (2012), pp. 3134-3162, https://doi.org/10.1137/090763093. · Zbl 1262.65178
[38] A. Mayo, Fast high order accurate solution of Laplace’s equation on irregular regions, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 144-157, https://doi.org/10.1137/0906012. · Zbl 0559.65082
[39] B. Müller, F. Kummer, and M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Internat. J. Numer. Methods Engrg., 96 (2013), pp. 512-528, https://doi.org/10.1002/nme.4569. · Zbl 1352.65083
[40] M. Oevermann and R. Klein, A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J. Comput. Phys., 219 (2006), pp. 749-769, https://doi.org/10.1016/j.jcp.2006.04.010. · Zbl 1143.35022
[41] C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), pp. 220-252, https://doi.org/10.1016/0021-9991(77)90100-0. · Zbl 0403.76100
[42] M. Petzoldt, Regularity results for Laplace interface problems in two dimensions, Z. Anal. Anwend., 20 (2001), pp. 431-455, https://doi.org/10.4171/ZAA/1024. · Zbl 1165.35333
[43] M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, New York, 1981. · Zbl 0453.41001
[44] J. A. Ro\u\itberg and Z. G. Šeftel, A homeomorphism theorem for elliptic systems, and its applications, Mat. Sb. (N.S.), 78 (120) (1969), pp. 446-472.
[45] C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes, PolyMesher: A general-purpose mesh generator for polygonal elements written in MATLAB, Struct. Multidiscip. Optim., 45 (2012), pp. 309-328, https://doi.org/10.1007/s00158-011-0706-z. · Zbl 1274.74401
[46] E. Wadbro, S. Zahedi, G. Kreiss, and M. Berggren, A uniformly well-conditioned, unfitted Nitsche method for interface problems, BIT, 53 (2013), pp. 791-820, https://doi.org/10.1007/s10543-012-0417-x. · Zbl 1279.65134
[47] F. Wang, Y. Xiao, and J. Xu, High-Order Extended Finite Element Methods for Solving Interface Problems, https://arxiv.org/abs/1604.06171, 2016. · Zbl 1442.74243
[48] L. Wang, S. Hou, and L. Shi, An improved non-traditional finite element formulation for solving three-dimensional elliptic interface problems, Comput. Math. Appl., 73 (2017), pp. 374-384, https://doi.org/10.1016/j.camwa.2016.11.035. · Zbl 1368.65237
[49] Q. Wang and J. Chen, An unfitted discontinuous Galerkin method for elliptic interface problems, J. Appl. Math., (2014), 241890, https://doi.org/10.1155/2014/241890. · Zbl 1406.65121
[50] X. S. Wang, L. Zhang, and W. K. Liu, On computational issues of immersed finite element methods, J. Comput. Phys., 228 (2009), pp. 2535-2551, https://doi.org/https://doi.org/10.1006/jcph.1996.5572. · Zbl 1158.74049
[51] Z. Wei, C. Li, and S. Zhao, A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems, Comput. Math. Appl., 75 (2018), pp. 2173-2192, https://doi.org/10.1016/j.camwa.2017.06.037. · Zbl 1409.65059
[52] H. Wu and Y. Xiao, An unfitted \(hp\)-interface penalty finite element method for elliptic interface problems, J. Comput. Math., 37 (2019), pp. 316-339. · Zbl 1449.65326
[53] J. J. Xu, Y. Xie, and B. Z. Lu, A parallel finite element solver for biomolecular simulations based on the toolbox PHG, J. Numer. Methods Comput. Appl., 37 (2016), pp. 67-82, https://lsec.cc.ac.cn/phg/download.htm. · Zbl 1374.92053
[54] S. Yu, Y. Zhou, and G. W. Wei, Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, J. Comput. Phys., 224 (2007), pp. 729-756, https://doi.org/10.1016/j.jcp.2006.10.030. · Zbl 1120.65333
[55] Y. C. Zhou and G. W. Wei, On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, J. Comput. Phys., 219 (2006), pp. 228-246, https://doi.org/10.1016/j.jcp.2006.03.027. · Zbl 1105.65108
[56] O. C. Zienkiewicz, R. L. Taylor, S. J. Sherwin, and J. Peiró, On discontinuous Galerkin methods, Internat. J. Numer. Methods Engrg., 58 (2003), pp. 1119-1148, https://doi.org/10.1002/nme.884. · Zbl 1032.76607
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