P. Angot, C. Bruneau, and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numerische Mathematik, vol.81, issue.4, pp.497-520, 1999.
DOI : 10.1007/s002110050401

E. Arquis and J. P. Caltagirone, Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieux poreux: application la convection naturelle, C.R. Acad. Sci. Paris II, vol.299, pp.1-4, 1984.

I. Babuska, Die Methode der finiten Elemente f??r elliptische Gleichungen mit diskontinuierlichen Koeffizienten, Computing, vol.19, issue.3, pp.207-213, 1970.
DOI : 10.1007/BF02248021

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp et al., PETSc users manual, 2008.

S. Balay, W. D. Gropp, L. C. Mcinnes, and B. F. Smith, Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries, Modern Software Tools in Scientific Computing, pp.163-202, 1997.
DOI : 10.1007/978-1-4612-1986-6_8

P. Bethelsen, A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, Journal of Computational Physics, vol.197, issue.1, pp.364-386, 2004.
DOI : 10.1016/j.jcp.2003.12.003

J. Bramble and J. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Advances in Computational Mathematics, vol.58, issue.1, pp.109-138, 1996.
DOI : 10.1007/BF02127700

D. Bresch, T. Colin, E. Grenier, B. Ribba, and O. Saut, Computational Modeling of Solid Tumor Growth: The Avascular Stage, SIAM Journal on Scientific Computing, vol.32, issue.4, 2009.
DOI : 10.1137/070708895
URL : https://hal.archives-ouvertes.fr/inria-00148610

F. Buret, N. Faure, L. Nicolas, R. Perussel, and C. Poignard, Numerical studies on the effect of electric pulses on an egg-shaped cell with a spherical nucleus, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00477495

F. Chantalat, C. H. Bruneau, C. Galusinski, and A. Iollo, Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design, Journal of Computational Physics, vol.228, issue.17, pp.6291-6315, 2009.
DOI : 10.1016/j.jcp.2009.05.017
URL : https://hal.archives-ouvertes.fr/hal-00385460

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik, vol.79, issue.2, pp.175-202, 1998.
DOI : 10.1007/s002110050336

I. Chern and Y. Shu, A coupling interface method for elliptic interface problems, Journal of Computational Physics, vol.225, issue.2, pp.2138-2174, 2007.
DOI : 10.1016/j.jcp.2007.03.012

R. E. Ewing, Z. Li, T. Lin, and Y. Lin, The immersed finite volume element methods for the elliptic interface problems, Mathematics and Computers in Simulation, vol.50, issue.1-4, pp.63-76, 1999.
DOI : 10.1016/S0378-4754(99)00061-0

R. P. Fedkiw, Coupling an Eulerian Fluid Calculation to a Lagrangian Solid Calculation with the Ghost Fluid Method, Journal of Computational Physics, vol.175, issue.1, pp.200-224, 2002.
DOI : 10.1006/jcph.2001.6935

R. P. Fedkiw, T. Aslam, B. Merriman, and S. Osher, A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method), Journal of Computational Physics, vol.152, issue.2
DOI : 10.1006/jcph.1999.6236

F. Gibou, R. P. Fedkiw, L. T. Cheng, and M. Kang, A Second-Order-Accurate Symmetric Discretization of the Poisson Equation on Irregular Domains, Journal of Computational Physics, vol.176, issue.1, pp.205-227, 2002.
DOI : 10.1006/jcph.2001.6977

F. Gibou and R. P. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, Journal of Computational Physics, vol.202, issue.2, pp.577-601, 2005.
DOI : 10.1016/j.jcp.2004.07.018

B. Gustafsson, A fourth order accurate discretization for the laplace and heat equations on arbitrary domains, with applications to the stefan problem, SIAM Journal of Numerical Analysis, vol.39, pp.396-406, 1975.

B. Gustafsson, The Convergence Rate for Difference Approximations to General Mixed Initial-Boundary Value Problems, SIAM Journal on Numerical Analysis, vol.18, issue.2, pp.179-190, 1981.
DOI : 10.1137/0718014

J. Huang and J. Zou, A mortar element method for elliptic problems with discontinuous coefficients. A mortar element method for elliptic problems with discontinuous coefficients, pp.549-576, 2002.

J. Huh and J. A. Sethian, Exact subgrid interface correction schemes for elliptic interface problems, Proceedings of the National Academy of Sciences of the United States of America, p.9874, 2008.
DOI : 10.1073/pnas.0707997105

H. Johansen and P. Colella, A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains, Journal of Computational Physics, vol.147, issue.1, pp.60-85, 1998.
DOI : 10.1006/jcph.1998.5965

R. J. Leveque and L. Z. Li, The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources, SIAM Journal on Numerical Analysis, vol.31, issue.4, pp.1019-1044, 1994.
DOI : 10.1137/0731054

Z. L. Li, A Fast Iterative Algorithm for Elliptic Interface Problems, SIAM Journal on Numerical Analysis, vol.35, issue.1, pp.230-254, 1998.
DOI : 10.1137/S0036142995291329

Z. L. Li and K. Ito, Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients, SIAM Journal on Scientific Computing, vol.23, issue.1, pp.339-361, 2001.
DOI : 10.1137/S1064827500370160

X. Liu, R. P. Fedkiw, and M. Kang, A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains, Journal of Computational Physics, vol.160, issue.1, pp.151-178, 2000.
DOI : 10.1006/jcph.2000.6444

B. Maury, A fat boundary method for the poisson problem in a domain with holes, Journal of Scientific Computing, vol.16, issue.3, pp.319-339, 2001.
DOI : 10.1023/A:1012821728631

A. Mayo, The Fast Solution of Poisson???s and the Biharmonic Equations on Irregular Regions, SIAM Journal on Numerical Analysis, vol.21, issue.2, pp.285-299, 1984.
DOI : 10.1137/0721021

A. Mayo, The rapid evaluation of volume integrals of potential theory on general regions, Journal of Computational Physics, vol.100, issue.2, pp.236-245, 1992.
DOI : 10.1016/0021-9991(92)90231-M

A. Mayo and A. Greenbaum, Fast Parallel Iterative Solution of Poisson???s and the Biharmonic Equations on Irregular Regions, SIAM Journal on Scientific and Statistical Computing, vol.13, issue.1, pp.101-118, 1992.
DOI : 10.1137/0913006

P. Mccorquodale, P. Collela, and H. Johansen, A Cartesian Grid Embedded Boundary Method for the Heat Equation on Irregular Domains, Journal of Computational Physics, vol.173, issue.2, pp.620-635, 2001.
DOI : 10.1006/jcph.2001.6900

M. Oevermann, C. Scharfenberg, and R. Klein, A sharp interface finite volume method for elliptic equations on Cartesian grids, Journal of Computational Physics, vol.228, issue.14, pp.5184-5206, 2009.
DOI : 10.1016/j.jcp.2009.04.018

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, 2003.
DOI : 10.1115/1.1760520
URL : http://dx.doi.org/10.1016/s0898-1221(03)90179-9

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, p.79, 1988.
DOI : 10.1016/0021-9991(88)90002-2

Y. Saad, Sparskit a basic tool-kit for sparse matrix computations

A. Sarthou, S. Vincent, P. Angot, and J. P. Caltagirone, The algebraic immersed interface and boundary method for elliptic equations with discontinuous coefficients, 2009.

J. A. Sethian, Level Set Methods and Fast Marching Methods, 1999.

J. A. Sethian, Evolution, Implementation, and Application of Level Set and Fast Marching Methods for Advancing Fronts, Journal of Computational Physics, vol.169, issue.2, pp.503-555, 2001.
DOI : 10.1006/jcph.2000.6657

M. Svard and J. Nordstrom, On the order of accuracy for difference approximations of initial-boundary value problems, Journal of Computational Physics, vol.218, issue.1, pp.333-352, 2006.
DOI : 10.1016/j.jcp.2006.02.014

A. Wiegmann and K. Bube, The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions, SIAM Journal on Numerical Analysis, vol.37, issue.3, pp.827-862, 2000.
DOI : 10.1137/S0036142997328664

S. Yu and G. W. Wei, Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities, Journal of Computational Physics, vol.227, issue.1, pp.602-632, 2007.
DOI : 10.1016/j.jcp.2007.08.003

X. Zhong, A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, Journal of Computational Physics, vol.225, issue.1, pp.1066-1099, 2007.
DOI : 10.1016/j.jcp.2007.01.017

Y. C. Zhou and G. W. Wei, On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, Journal of Computational Physics, vol.219, issue.1, pp.228-246, 2006.
DOI : 10.1016/j.jcp.2006.03.027

Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, Journal of Computational Physics, vol.213, issue.1, pp.1-30, 2006.
DOI : 10.1016/j.jcp.2005.07.022

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