A POSTERIORI ERROR ANALYSIS FOR POISSON'S EQUATION APPROXIMATED BY XFEM

This paper presents and studies a residual a posteriori error estimator for Laplace's equa- tion in two space dimensions approximated by the eXtended Finite Element Method (XFEM). The XFEM allows to perform finite element computations on multi-cracked domains by using meshes of the non-cracked domain. The main idea consists of adding supplementary basis functions of Heaviside type and singular functions in order to take into account the crack geometry and the singularity at the crack tip respectively. Resume. Dans ce travail on propose et onetudie un estimateur d'erreur par residu pour l'´equation de Laplace en deux dimensions d'espace discretisee par la methode d'´elements finis ´ (XFEM). La XFEM permet de realiser des simulations parements finis sur des domaines multi-fissures en utilisant des maillages du domaine non fissure. L'idee principale de la methode consisteajouter des fonctions de base supplementaires de type Heaviside et des fonctions singulieres afin de prendre en compte la geometrie de la fissure et la singularite en pointe de fissure.

[1]  Jean-Luc Guermond,et al.  Eléments finis : théorie, applications, mise en œuvre , 2002 .

[2]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[3]  Ricardo H. Nochetto,et al.  Residual type a posteriori error estimates for elliptic obstacle problems , 2000, Numerische Mathematik.

[4]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[5]  V. Girault,et al.  A Local Regularization Operator for Triangular and Quadrilateral Finite Elements , 1998 .

[6]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[7]  Nicolas Moës,et al.  X-FEM, de nouvelles frontières pour les éléments finis , 2002 .

[8]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[9]  S. Hilbert,et al.  A Mollifier Useful for Approximations in Sobolev Spaces and Some Applications to Approximating Solutions of Differential Equations , 1973 .

[10]  Jaroslav Haslinger,et al.  A New Fictitious Domain Approach Inspired by the Extended Finite Element Method , 2009, SIAM J. Numer. Anal..

[11]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[12]  Patrick Laborde,et al.  Numerical Analysis A quasi-optimal convergence result for fracture mechanics with XFEM , 2006 .

[13]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[14]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[15]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[16]  H. Schönheinz.  G. Strang / G. J. Fix, An Analysis of the Finite Element Method. (Series in Automatic Computation. XIV + 306 S. m. Fig. Englewood Clifs, N. J. 1973. Prentice‐Hall, Inc. , 1975 .

[17]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[18]  Gilbert Strang,et al.  Approximation in the finite element method , 1972 .