zbMATH — the first resource for mathematics

Analysis of an efficient finite element method for embedded interface problems. (English) Zbl 1190.65172
Summary: A stabilized finite element method based on the Nitsche technique for enforcing constraints leads to an efficient computational procedure for embedded interface problems, in which the finite element mesh need not be aligned with the interface geometry. We consider cases in which the jump of a field across the interface is given, as well as cases in which the primary field on the interface is given. Optimal rates of convergence hold. Representative numerical examples demonstrate the effectiveness of the proposed methodology.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76F10 Shear flows and turbulence
76M10 Finite element methods applied to problems in fluid mechanics
Full Text: DOI
[1] Chessa J, Smolinski P, Belytschko T (2000) The extended finite element method for solidification problems. Int J Numer Methods Eng 28(5): 339–350 · Zbl 1003.80004
[2] Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48(12): 1741–1760 · Zbl 0989.74066
[3] Dolbow J, Harari I (2009) An efficient finite element method for embedded interface problems. Internat. J Numer Methods Eng 78(2): 229–252. doi: 10.1002/nme.2486 · Zbl 1183.76803
[4] Dolbow J, Mosso S, Robbins J, Voth T (2008) Coupling volume-of-fluid based interface reconstructions with the extended finite element method. Comput Methods Appl Mech Eng 197: 439–447 · Zbl 1169.76409
[5] Dolbow JE, Franca L (2008) Residual-free bubbles for embedded Dirichlet problems. Comput Methods Appl Mech Eng 197: 3751–3759 · Zbl 1197.65180
[6] Dolbow JE, Fried E, Ji H (2005) A numerical strategy for investigating the kinetic response of stimulus responsive hydrogels. Comput Methods Appl Mech Eng 194: 4447–4480 · Zbl 1094.76039
[7] Duddu R, Bordas S, Chopp D, Moran B (2008) A combinded extended finite element and level set method for biofilm growth. Int J Numer Methods Eng 74(5): 848–870 · Zbl 1195.74169
[8] Elias R, Martins M, Coutinho A (2007) Simple finite element-based computation of distance functions in unstructured grids. Int J Numer Methods Eng 72(9): 1095–1110 · Zbl 1194.65145
[9] Fedkiw R, Aslam T, Merriman B, Osher S (1999) A non- oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J Comput Phys 152(2): 457–492 · Zbl 0957.76052
[10] Fries T (2009) The intrinsic xfem for two-fluid flows. Int J Numer Methods Eng 60: 437–471 · Zbl 1161.76026
[11] Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191(47–48): 5537–5552. doi: 10.1016/S0045-7825(02)00524-8 · Zbl 1035.65125
[12] Ji H, Dolbow JE (2004) On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. Int J Numer Methods Eng 61(14): 2508–2535 · Zbl 1075.74651
[13] Juntunen M, Stenberg R (2009) Nitsche’s method for general boundary conditions. Math Comput 78(267): 1353–1374. doi: 10.1090/S0025-5718-08-02183-2 · Zbl 1198.65223
[14] Liu X, Fedkiw R, Kang M (2000) A boundary condition capturing method for Poisson’s equation on irregular domains. J Comput Phys 160: 151–178 · Zbl 0958.65105
[15] Merle R, Dolbow J (2002) Solving thermal and phase change problems with the extended finite element method. Comptu Mech 28(5): 339–350 · Zbl 1073.76589
[16] Moës N, Bechet E, Tourbier M (2006) Imposing essential boundary conditions in the extended finite element method. Int J Numer Methods Eng 67(12): 1641–1669 · Zbl 1113.74072
[17] Moës N, Cloirec M, Cartraud P, Remacle J (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192(3): 3163–3177 · Zbl 1054.74056
[18] Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1): 131–150. doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J · Zbl 0955.74066
[19] Mourad H, Dolbow J, Garikipati K (2005) An assumed-gradient finite element method for the level-set equation. Int J Numer Methods Eng 8(8): 1009–1032 · Zbl 1114.65115
[20] Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh Math Sem Univ Hamburg 36(1): 9–15. doi: 10.1007/BF02995904 · Zbl 0229.65079
[21] Song JH, Areias PMA, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Methods Eng 67(6): 868–868. doi: 10.1002/nme.1652 · Zbl 1113.74078
[22] Stenberg R (1995) On some techniques for approximating boundary conditions in the finite element method. J Comput Appl Math 63: 139–148 · Zbl 0856.65130
[23] Tezduyar TE, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces–the deforming-spatial-domain/space-time procedure. I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94(3): 339–351. doi: 10.1016/0045-7825(92)90059-S · Zbl 0745.76044
[24] Valance S, de Borst R, Rethore J, Coret M (2008) A partition-of-unity-based finite element method for level sets. Int J Numer Methods Eng 76(10): 1513–1527 · Zbl 1195.65134
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.