A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources

We compare the Immersed Interface Method (IIM) with the Extended Finite Element Method (X-FEM) for elliptic equations with singular sources and discontinuous coefficients. The IIM has been compared favorably with a number of other competing methods. These methods are of particular interest because they allow for the solution of elliptic equations with internal boundaries on nonconforming meshes. In the context of moving interface problems, the emphasis in this paper is placed on accuracy of solutions and their normal derivatives on the interface. These methods are briefly described and the results for benchmark problems are compared.

[1]  D. Chopp,et al.  Extended finite element method and fast marching method for three-dimensional fatigue crack propagation , 2003 .

[2]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[3]  David L. Chopp,et al.  A hybrid extended finite element/level set method for modeling phase transformations , 2002 .

[4]  Ted Belytschko,et al.  Discontinuous enrichment in finite elements with a partition of unity method , 2000 .

[5]  John E. Dolbow,et al.  On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method , 2004 .

[6]  Ted Belytschko,et al.  The extended finite element method for rigid particles in Stokes flow , 2001 .

[7]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[8]  T. Belytschko,et al.  Extended finite element method for three-dimensional crack modelling , 2000 .

[9]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

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

[11]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[12]  Zhilin Li.  AN OVERVIEW OF THE IMMERSED INTERFACE METHOD AND ITS APPLICATIONS , 2003 .

[13]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[14]  Ted Belytschko,et al.  An extended finite element method for modeling crack growth with frictional contact , 2001 .

[15]  T. Belytschko,et al.  The extended finite element method (XFEM) for solidification problems , 2002 .

[16]  Ted Belytschko,et al.  Modelling crack growth by level sets in the extended finite element method , 2001 .

[17]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[18]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .