A new fictitious domain approach inspired by the extended finite element method

SIAM Journal on Numerical Analysis

Volumn 47, Issue 2, 2009, Pages 1474-1499

  Haslinger, Jaroslav ^a   Renard, Yves ^b  

^a CHARLES UNIVERSITY   (Czech Republic)

^b UNIVERSITÉ LYON 1   (France)

Author keywords

Approximation of elliptic problems; Fictitious domain; Stabilization technique; Xfem

Indexed keywords

A-PRIORI ESTIMATES; ELLIPTIC PROBLEM; EXTENDED FINITE ELEMENT METHOD; FICTITIOUS DOMAINS; NEUMANN AND DIRICHLET BOUNDARY CONDITIONS; NUMERICAL EXPERIMENTS; OPTIMAL METHODS; POISSON PROBLEM; STABILIZATION TECHNIQUES; XFEM;

BOUNDARY CONDITIONS; POISSON EQUATION; STABILIZATION;

NUMERICAL METHODS;

EID: 76749153485     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070704435     Document Type: Article

References (28)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. H. J. C. BARBOSA AND T. J. R. HUGHES, The finite element method with Lagrange multipliers on the boundary: Circumventing the Babuska-Brezzi condition, Comput. Methods Appl. Mech. Engrg., 85(1991), pp. 109-128.
3. H. J. C. BARBOSA AND T. J. R. HUGHES, Boundary Lagrange multipliers infinite element methods: Error analysis in natural norms, Numer. Math., 62(1992), pp. 1-15.
4. P. HANSBO, C. LOVADINA, I. PERUGIA, AND G. SANGALLI, A Lagrange multiplier method for the finite element solution of elliptic interface problems using nonmatching meshes, Numer. Math., 100(2005), pp. 91-115.
5. E. CHAHINE, P. LABORDE, AND Y. RENARD, Crack-tip enrichment in the Xfem method using a cut-off function, Internat. J. Numer. Methods Engrg., 75(2008), pp. 629-646.
6. P. G. CIARLET, The finite element method for elliptic problems, Studies in Mathematics and its Applications 4, North-Holland, Amsterdam, 1978.
7. P. G. CIARLET, The finite element method for elliptic problems, in Handbook of Numerical Analysis, Volume II, Part 1, P. G. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 1991, pp. 17-352.
8. P. CLÉMENT, Approximation by finite elements functions using local regularization, RAIRO, Anal. Numer., 9(1975), pp. 77-84.
9. J. W. DEMMEL, J. R. GILBERT, AND X. S. LI, A general purpose library for the direct solution of large, sparse, nonsymmetric systems, http://crd.lbl.gov/ ~xiaoye/SuperLU/.
10. A. ERN AND J.-L. GUERMOND, Theory and practice of finite elements, Appl. Math. Sci., 159 (2004).
11. V. GIRAULT AND R. GLOWINSKI, Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan J. Indust. Appl. Math., 12(1995), pp. 487-514.
12. P. GRISVARD, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.
13. J. HASLINGER AND A. KLARBRING, Fictitious domain/mixed finite element approach for a class of optimal shape design problems, Math. Model. Numer. Anal. (M2AN), 4(1995), pp. 435-450.
14. P. HILD AND Y. RENARD, A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics, submitted.
15. T. HUGHES AND L. P. FRANCA, A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65(1987), pp. 85-96.
16. P. LABORDE, J. POMMIER, Y. RENARD, AND M. SALAÜN, High order extended finite element method for cracked domains, Internat. J. Numer. Methods Engng., 64(2005), pp. 354-381.
17. N. MOËS, E. BÉCHET, AND M. TOURBIER, Imposing Dirichlet boundary conditions in the eX-tended Finite Element Method, Internat. J. Numer. Methods Engng., 12(2006), pp. 354-381.
18. N. MOËS, J. DOLBOW, AND T. BELYTSCHKO, A finite element method for crack growth without remeshing, Internat. J. Numer. Methods Engrg., 46(1999), pp. 131-150.
19. N. MOËS, A. GRAVOUIL, AND T. BELYTSCHKO, Non-planar 3D crack growth by the extended finite element and level sets, Part I: Mechanical model, Internat. J. Numer. Methods Engrg., 11(2002), pp. 2549-2568.
20. J. NITSCHE, Uber ein Variationsprinzip zur Losung von Dirichlet-Problemen bei Verwendung von Teilraumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Univ. Hamburg, 36(1971), pp. 9-15.
21. J. PITKÄRANTA, Local stability conditions for the Babuska method of Lagrange multipliers, Math. Comput., 35(1980), pp. 1113-1129.
22. Y. RENARD AND J. POMMIER, Getfem++. An open source generic C++ library for finite element methods, http://home.gna.org/getfem/.
23. F. L. STAZI, E. BUDYN, J. CHESSA, AND T. BELYTSCHKO, An extended finite element method with higher-order elements for curved cracks, Comput. Mech., 31(2003), pp. 38-48.
24. R. STENBERG, On some techniques for approximating boundary conditions in the finite element method, J. Comput. Appl. Math., 63(1995), pp. 139-148.
25. M. STOLARSKA, D. L. CHOPP, N. MOËS, AND T. BELYTSCHKO, Modelling crack growth by level sets in the extended finite element method, Internat. J. Numer. Methods Engrg., 51(2001), pp. 943-960.
26. G. STRANG AND G. J. FIX, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
27. N. SUKUMAR, D. L. CHOPP, N. MOËS, AND T. BELYTSCHKO, Modeling holes and inclusions by level sets in the extended finite element method, Comput. Methods Appl. Mech. Eng., 46(2001), pp. 6183-6200.
28. N. SUKUMAR, N. MOËS, B. MORAN, AND T. BELYTSCHKO, Extended finite element method for three dimensional crack modelling, Internat. J. Numer. Methods Engrg., 48(2000), pp. 1549-1570.