A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Journal of Computational Physics

Volumn 231, Issue 4, 2012, Pages 2015-2048

  Hellrung, Jeffrey Lee ^a   Wang, Luming ^b   Sifakis, Eftychios ^c   Teran, Joseph M ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c University of Wisconsin   (United States)

Author keywords

Elliptic interface problems; Embedded interface methods; Multigrid methods; Variational methods; Virtual node methods

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); LINEAR SYSTEMS; NUMERICAL METHODS; POISSON EQUATION;

DISCRETIZATIONS; ELLIPTIC INTERFACE PROBLEMS; EMBEDDED INTERFACE; EMBEDDED INTERFACE METHOD; IRREGULAR DOMAINS; MULTIGRID METHODS; THREE DIMENSIONS; VARIATIONAL METHODS; VIRTUAL NODE; VIRTUAL NODE METHOD;

LAGRANGE MULTIPLIERS;

EID: 84855188577     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.11.023     Document Type: Article

References (93)

1. Bedrossian J., von Brecht J.H., Zhu S., Sifakis E., Teran J.M. A second order virtual node method for elliptic problems with interfaces and irregular domains. J. Comput. Phys. 2010, 229(18):6405-6426.
2. BabuŠka I. The finite element method for elliptic equations with discontinuous coefficients. Computing 1970, 5:207-213.
3. Bramble J., King J. A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 1996, 6:109-138.
4. Chen Z., Zou J. Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math 1996, 79:175-202.
5. Dryja M. A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients. Numer. Math. 2005, 99:645-656.
6. Cockburn B., Gopalakrishnan J., Lazarov R. Unified hybridization of discontinuous Galerkin, mixed, and conforming Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 2009, 47:1319-1365.
7. Wohlmuth B.I., Krause R.H. Multigrid methods based on the unconstrained product space arising from mortar finite element discretizations. SIAM J. Numer. Anal. 1999, 39:2001.
8. Huang J., Zou J. A mortar element method for elliptic problems with discontinuous coefficients. IMA J. Numer. Anal. 2001, 22(4):549-576.
9. Lamichhane B.P., Wohlmuth B.I. Mortar finite elements for interface problems. Computing 2004, 72:333-348.
10. Leveque R.J., Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 1994, 31:1019-1044.
11. LeVeque R.J., Li Z. Immersed interface methods for stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 1997, 18:709-735.
12. Li Z., Lai M.-C. The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys. 2001, 171:822-842.
13. Lee L., LeVeque R.J. An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 2003, 25:832-856.
14. Le D.V., Khoo B.C., Peraire J. An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries. J. Comput. Phys. 2006, 220:109-138.
15. Xu S., Wang Z.J. An immersed interface method for simulating the interaction of a fluid with moving boundaries. J. Comput. Phys. 2006, 216:454-493.
16. Tan Z., Le D.V., Li Z., Lim K.M., Khoo B.C. An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane. J. Comput. Phys. 2008, 227:9955-9983.
17. Xu S., Wang Z.J. A 3d immersed interface method for fluid-solid interaction. Comput. Methods Appl. Mech. Eng. 2008, 197(25-28):2068-2086. Immersed Boundary Method and Its Extensions.
18. Li Z., Ito K. The immersed interface method: numerical solutions of pdes involving interfaces and irregular domains. Frontiers in Applied Mathematics 2006, vol. 33. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
19. Beale J.T., Layton A.T. On the accuracy of finite difference methods for elliptic problems with interfaces. Commun. Appl. Math. Comput. Sci. 2006, 1:91-119.
20. Li Z., Ito K. Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 2001, 23:339-361.
21. Deng S., Ito K., Li Z. Three-dimensional elliptic solvers for interface problems and applications. J. Comput. Phys. 2003, 184:215-243.
22. Chen T., Strain J. Piecewise-polynomial discretization and krylov-accelerated multigrid for elliptic interface problems. J. Comput. Phys. 2008, 227:7503-7542.
23. Li Z. A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 1998, 35:230-254.
24. Wiegmann A., Bube K.P. The explicit-jump immersed interface method: finite difference methods for pdes with piecewise smooth solutions. SIAM J. Numer. Anal. 2000, 37:827-862.
25. Berthelsen P.A. A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys. 2004, 197:364-386.
26. Adams L., Li Z. The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 2002, 24:463-479.
27. Adams L., Chartier T.P. New geometric immersed interface multigrid solvers. SIAM J. Sci. Comput. 2004, 25:1516-1533.
28. Adams L., Chartier T.P. A comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems. SIAM J. Sci. Comput. 2005, 26:762-784.
29. Liu X.-D., Fedkiw R.P., Kang M. A boundary condition capturing method for poisson's equation on irregular domains. J. Comput. Phys. 2000, 160:151-178.
30. Gibou F., Fedkiw R.P., Cheng L.-T., Kang M. A second-order-accurate symmetric discretization of the poisson equation on irregular domains. J. Comput. Phys. 2002, 176:205-227.
31. Gibou F., Fedkiw R. A fourth order accurate discretization for the laplace and heat equations on arbitrary domains, with applications to the stefan problem. J. Comput. Phys. 2005, 202:577-601.
32. Jomaa Z., Macaskill C. The embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions. J. Comput. Phys. 2005, 202:488-506.
33. Zhou Y.C., Zhao S., Feig M., Wei G.W. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 2006, 213:1-30.
34. Chern I.-L., Shu Y.-C. A coupling interface method for elliptic interface problems. J. Comput. Phys. 2007, 225:2138-2174.
35. Liu X.-D., Sideris T.C. Convergence of the ghost fluid method for elliptic equations with interfaces. Math. Comput. 2003, 72:1731-1746.
36. Molino N., Bao Z., Fedkiw R. A virtual node algorithm for changing mesh topology during simulation. ACM SIGGRAPH 2005 Courses, SIGGRAPH '05 2005, ACM, New York, NY, USA.
37. Bao Z., Hong J.-M., Teran J., Fedkiw R. Fracturing rigid materials. IEEE Trans. Visual. Comput. Graph. 2007, 13:370-378.
38. E. Sifakis, K.G. Der, R. Fedkiw, Arbitrary cutting of deformable tetrahedralized objects, in: Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on Computer animation, SCA '07, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 2007, pp. 73-80.
39. Zhou Y.C., Wei G.W. On the fictitious-domain and interpolation formulations of the matched interface and boundary (mib) method. J. Comput. Phys. 2006, 219:228-246.
40. Yu S., Wei G.W. Three-dimensional matched interface and boundary (mib) method for treating geometric singularities. J. Comput. Phys. 2007, 227:602-632.
41. Zhao S., Wei G.W. Matched interface and boundary (mib) for the implementation of boundary conditions in high-order central finite differences. Int. J. Numer. Meth. Eng. 2009, 77:1690-1730.
42. Pan K., Tan Y., Hu H. An interpolation matched interface and boundary method for elliptic interface problems. J. Comput. Appl. Math. 2010, 234:73-94.
43. Hou S., Liu X.-D. A numerical method for solving variable coefficient elliptic equation with interfaces. J. Comput. Phys. 2005, 202:411-445.
44. Hou S., Wang W., Wang L. Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces. J. Comput. Phys. 2010, 229:7162-7179.
45. Ng Y.T., Min C., Gibou F. An efficient fluid-solid coupling algorithm for single-phase flows. J. Comput. Phys. 2009, 228:8807-8829.
46. Papac J., Gibou F., Ratsch C. Efficient symmetric discretization for the poisson, heat and stefan-type problems with robin boundary conditions. J. Comput. Phys. 2010, 229:875-889.
47. Glowinski R., Pan T.-W., Periaux J. A fictitious domain method for dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 1994, 111:283-303.
48. Almgren A.S., Bell J.B., Colella P., Marthaler T. A cartesian grid projection method for the incompressible euler equations in complex geometries. SIAM J. Sci. Comput. 1997, 18:1289-1309.
49. Parussini L., Pediroda V. Fictitious domain approach with hp-finite element approximation for incompressible fluid flow. J. Comput. Phys. 2009, 228:3891-3910.
50. Jomaa Z., Macaskill C. The shortley-weller embedded finite-difference method for the 3d poisson equation with mixed boundary conditions. J. Comput. Phys. 2010, 229:3675-3690.
51. Lew A.J., Buscaglia G.C. A discontinuous-galerkin-based immersed boundary method. Int. J. Numer. Methods Eng. 2008, 76:427-454.
52. Guyomarc'h G., Lee C.-O., Jeon K. A discontinuous galerkin method for elliptic interface problems with application to electroporation. Commun. Numer. Methods Eng. 2009, 25:991-1008.
53. Moës N., Dolbow J., Belytschko T. A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 1999, 46:131-150.
54. Daux C., Moës N., Dolbow J., Sukumar N., Belytschko T. Arbitrary branched and intersecting cracks with the extended finite element method. Int. J. Numer. Meth. Eng. 2000, 48:1741-1760.
55. Belytschko T., Moës N., Usui S., Parimi C. Arbitrary discontinuities in finite elements. Int. J. Numer. Methods Eng. 2001, 50:993-1013.
56. Moës N., Cloirec M., Cartraud P., Remacle J.F. A computational approach to handle complex microstructure geometries. Comput. Methods Appl. Mech. Eng. 2003, 192(28-30):3163-3177.
57. Ji H., Dolbow J.E. On strategies for enforcing interfacial constraints and evaluating jump conditions with extended finite element method. Int. J. Numer. Meth. Eng. 2004, 61:2508-2535.
58. Fries T.-P., Belytschko T. The intrinsic xfem: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Methods Eng. 2006, 68:1358-1385.
59. Moës N., Béchet E., Tourbier M. Imposing dirichlet boundary conditions in the extended finite element method. Int. J. Numer. Meth. Eng. 2006, 67:1641-1669.
60. Groi{dotless}́ S., Reusken A. An extended pressure finite element space for two-phase incompressible flows with surface tension. J. Comput. Phys. 2007, 224:40-58.
61. van der Bos F., Gravemeier V. Numerical simulation of premixed combustion using an enriched finite element method. J. Comput. Phys. 2009, 228:3605-3624.
62. B.L. Vaughan, Jr., B.G. Smith, D.L. Chopp, A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources, Commun. Appl. Math. Comput. Sci. 1 (1) (2006) 207-228.
63. Beale J.T., Chopp D.L., LeVeque R.J., Li Z. Correction to the article a comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources by Vaughan et al. Commun. Appl. Math. Comput. Sci. 2008, 3(1).
64. Young D.P., Melvin R.G., Bieterman M.B., Johnson F.T., Samant S.S., Bussoletti J.E. A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics. J. Comput. Phys. 1990, 92:1-66.
65. Hansbo A., Hansbo P. An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 2002, 191:5537-5552.
66. Li Z., Lin T., Wu X. New cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 2003, 96(1):61-98.
67. Hansbo A., Hansbo P. A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 2004, 193:3523-3540.
68. Song J.-H., Areias P.M.A., Belytschko T. A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Methods Eng. 2006, 67:868-893.
69. Mourad H.M., Dolbow J., Harari I. A bubble-stabilized finite element method for dirichlet constraints on embedded interfaces. Int. J. Numer. Methods Eng. 2007, 69:772-793.
70. Dolbow J., Franca L. Residual-free bubbles for embedded dirichlet problems. Comput. Methods Appl. Mech. Eng. 2008, 197:3751-3759.
71. Kumar A.V., Padmanabhan S., Burla R. Implicit boundary method for finite element analysis using non-conforming mesh or grid. Int. J. Numer. Meth. Eng. 2008, 74:1421-1447.
72. Dolbow J., Harari I. An efficient finite element method for embedded interface problems. Int. J. Numer. Methods Eng. 2009, 78:229-252.
73. Harari I., Dolbow J. Analysis of an efficient finite element method for embedded interface problems. Comput. Mech. 2010, 46:205-211.
74. 1-nonconforming finite element method for interface problems. SIAM J. Numer. Anal. 2010, 48(6):2117-2134.
75. H. Wu, Y. Xiao, An unfitted hp-interface penalty finite element method for elliptic interface problemsarXiv:arXiv/1007.2893.
76. Richardson C.L., Hegemann J., Sifakis E., Hellrung J., Teran J.M. An XFEM method for modeling geometrically elaborate crack propagation in brittle materials. Int. J. Numer. Meth. Eng. 2011, 88(10):1042-1065.
77. BabuŠka I. The finite element method with lagrangian multipliers. Numer. Math. 1973, 20:179-192.
78. Pitkäranta J. Boundary subspaces for the finite element method with lagrange multipliers. Numer. Math. 1979, 33:273-289.
79. Chapelle D., Bathe K. The inf-sup test. Comput. Struct. 1993, 47(4-5):537-545.
80. Li Z. The immersed interface method using a finite element formulation. Appl. Numer. Math. 1998, 27:253-267.
81. Johansen H., Colella P. A cartesian grid embedded boundary method for poisson's equation on irregular domains. J. Comput. Phys. 1998, 147:60-85.
82. Schwartz P., Barad M., Colella P., Ligocki T. A cartesian grid embedded boundary method for the heat equation and poisson's equation in three dimensions. J. Comput. Phys. 2006, 211:531-550.
83. R.K. Crockett, P. Colella, D.T. Graves, A cartesian grid embedded boundary method for solving the poisson and heat equations with discontinuous coefficients in three dimensionsLBNL Paper LBNL-2929E.
84. Oevermann M., Klein R. A cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J. Comput. Phys. 2006, 219:749-769.
85. Oevermann M., Scharfenberg C., Klein R. A sharp interface finite volume method for elliptic equations on cartesian grids. J. Comput. Phys. 2009, 228:5184-5206.
86. McAdams A., Sifakis E., Teran J. A parallel multigrid poisson solver for fluids simulation on large grids. ACM SIGGRAPH/Eurograph Symp. Comput. Anim. 2010, 1-10.
87. Wan J.W.L., Liu X.-D. A boundary condition-capturing multigrid approach to irregular boundary problems. SIAM J. Sci. Comput. 2004, 25:1982-2003.
88. Min C., Gibou F. Geometric integration over irregular domains with application to level-set methods. J. Comput. Phys. 2007, 226:1432-1443.
89. Benzi M., Golub G.H., Liesen J. Numerical solution of saddle point problems. Acta Numer. 2005, 14:1-137.
90. S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, Petsc, 2009. web page, http://www.mcs.anl.gov/petsc.
91. S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, Petsc users manual, Tech. Rep. ANL-95/11-Revision 3.0.0, Argonne National Laboratory, 2008.
92. S. Balay, W.D. Gropp, L.C. McInnes, B.F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in: E. Arge, A.M. Bruaset, H.P. Langtangen (Eds.), Modern Software Tools in Scientific Computing, Birkh, pp. 163-202.
93. Cowper G.R. Gaussian quadrature formulas for triangles. Int. J. Numer. Methods Eng. 1973, 7:405-408.