Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T14:21:49.863Z Has data issue: false hasContentIssue false
  • English
  • Français

A Parallel Second Order Cartesian Method for Elliptic Interface Problems

Published online by Cambridge University Press:  20 August 2015

Marco Cisternino  and
Lisl Weynans
Marco Cisternino*
Affiliation:
Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino C.so Duca degli Abruzzi 24, 10129 Torino, Italy
Lisl Weynans*
Affiliation:
Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France. CNRS, IMB, UMR 5251, F-33400 Talence, France. INRIA, F-33400 Talence, France
*
Corresponding author.Email address:lisl.weynans@math.u-bordeaux1.fr
Email address:marco.cisternino@polito.it
Get access

Abstract

We present a parallel Cartesian method to solve elliptic problems with complex immersed interfaces. This method is based on a finite-difference scheme and is second-order accurate in the whole domain. The originality of the method lies in the use of additional unknowns located on the interface, allowing to express straightforwardly the interface transmission conditions. We describe the method and the details of its parallelization performed with the PETSc library. Then we present numerical validations in two dimensions, assorted with comparisons to other related methods, and a numerical study of the parallelized method.

Keywords

65N0665N1265Y05Elliptic interface problemCartesian methodsecond-order schemeinterface unknowns
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1562 - 1587
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] The Message Passing Interface Forum Home Page. http://www.mpi-forum.org.Google Scholar
[2] Plateforme Federative pour la Recherche en Informatique et Mathematiques en Aquitaine. https://plafrim.bordeaux.inria.fr.Google Scholar
[3]Angot, P., Bruneau, C.-H., and Fabrie, P.. A penalization method to take into account obstacles in incompressible flows. Numer. Math., 81(4):497–520,1999.Google Scholar
[4]Arquis, E. and Caltagirone, J.P.. Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieux poreux: application la convection naturelle. C.R. Acad. Sci. Paris II, 299:1–4,1984.Google Scholar
[5]Babuska, I.. The finite element method for elliptic equations with discontinuous coefficients. Computing, 5:207–213,1970.Google Scholar
[6]Balay, Satish, Brown, Jed, Buschelman, Kris, Eijkhout, Victor, Gropp, William D., Kaushik, Dinesh, G. Knepley, Matthew, McInnes, Lois Curfman, Smith, Barry F., and Zhang, Hong. PETSc users manual. Technical Report ANL-95/11 - Revision 3.0.0, Argonne National Laboratory, 2008.Google Scholar
[7]Balay, Satish, Brown, Jed, Buschelman, Kris, Gropp, William D., Kaushik, Dinesh, Knepley, Matthew G., McInnes, Lois Curfman, Smith, Barry F., and Zhang, Hong. PETSc Web page, 2009. http://www.mcs.anl.gov/petsc.Google Scholar
[8]Balay, Satish, Gropp, William D., McInnes, Lois Curfman, and Smith, Barry F.. Efficient management of parallelism in object oriented numerical software libraries. In Arge, E., Bruaset, A. M., and Langtangen, H. P., editors, Modern Software Tools in Scientific Computing, pages 163–202. Birkhauser Press, 1997.Google Scholar
[9]Bethelsen, P.. A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys., 197:364–386, 2004.Google Scholar
[10]Bramble, J. and King, J.. A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math., 6:109–138,1996.Google Scholar
[11]Bresch, D., Colin, T., Grenier, E., Ribba, B., and Saut, O.. Computational modeling of solid tumor growth: the avascular stage, 2009.Google Scholar
[12]Buret, F., Faure, N., Nicolas, L., Perussel, R., and Poignard, C.. Numerical studies on the effect of electric pulses on an egg-shaped cell with a spherical nucleus. Technical Report 7270, INRIA, 2010.Google Scholar
[13]Chen, Z. and Zou, J.. Finite element methods and their convergence for elliptic and parabolic interface problems,. Numer. Math., 79:175–202,1998.Google Scholar
[14]Chern, I. and Shu, Y.-C.. A coupling interface method for elliptic interface problems. J. Comput. Phys., 225:2138–2174,2007.Google Scholar
[15]Ewing, R.E., Li, Z., Lin, T., and Lin, Y.. The immersed finite volume elements methods for the elliptic interface problems. Mathematics and Computers in Simulation, 50:63–76,1999.Google Scholar
[16]Fedkiw, R. P.. Coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method. J. Comput. Phys., 175:200–224,2002.Google Scholar
[17]Fedkiw, R. P., Aslam, T., Merriman, B., and Osher, S.. A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys., 152:457–492, 1999.Google Scholar
[18]Gibou, F., Fedkiw, R. P., Cheng, L.T., and Kang, M.. A second order accurate symmetric discretization of the poisson equation on irregular domains. J. Comput. Phys., 176:205–227, 2002.Google Scholar
[19]Gibou, F. and Fedkiw, R.P.. A fourth order accurate discretization for the laplace and heat equations on arbitrary domains, with applications to the stefan problem. J. Comput. Phys., 202:577–601,2005.CrossRefGoogle Scholar
[20]Gustafsson, B.. 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, 39:396–406,1975.Google Scholar
[21]Gustafsson, B.. The convergence rate for difference approximations to general initial boundary value problems. SIAM Journal of Numerical Analysis, 18:179–190,1981.CrossRefGoogle Scholar
[22]Huang, J. and Zou, J.. A mortar element method for elliptic problems with discontinuous coefficients. A mortar element method for elliptic problems with discontinuous coefficients, 22:549–576, 2002.Google Scholar
[23]Huh, J.-S. and Sethian, J.A.. Exact subgrid interface correction schemes for elliptic interface problems. Proceedings of the National Academy of Sciences of the United States of America, 105:9874,2008.Google Scholar
[24]Johansen, H. and Colella, P.. A cartesian grid embedded boundary method for poisson's equation on irregular domains. J. Comput. Phys., 147:60–85,1998.CrossRefGoogle Scholar
[25]Leveque, R. J. and Li, L.Z.. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Numerical Analysis, 31(4):1019–1044, 1994.Google Scholar
[26]Li, Z.L.. A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal., 35:230–254,1998.Google Scholar
[27]Li, Z.L. and Ito, K.. Maximum principle preserving schemes for interfacxe problems with discontinuous coefficients. SIAM J. Sci. Comput., 23:339–361,2001.Google Scholar
[28]Liu, X.-D., Fedkiw, R. P., and Kang, M.. A boundary capturing method for poisson's equation on irregular domains. J. Comput. Phys., 160:151–178,2000.Google Scholar
[29]Maury, B.. A fat boundary method for the poisson problem in a domain with holes. Journal of Scientific Computing, 16:319–339,2001.Google Scholar
[30]Mayo, A.. The fast solution of poisson's and the biharmonic equations on general regions. SIAM J. Numer. Anal., 21:285–299,1984.CrossRefGoogle Scholar
[31]Mayo, A.. The rapid evaluation of volume integrals of potential theory on general regions. J. Comput. Phys., 100:236–245,1992.Google Scholar
[32]Mayo, A. and Greenbaum, A.. Fast parallel iterative solution of poisson's and the biharmonic equations on irregular regions. SIAM J. Sci. Stat. Comput., 13:101–118,1992.Google Scholar
[33]McCorquodale, P., Collela, P., and Johansen, H.. A cartesian grid embedded boundary method for the heat equation on irregular domains. J. Comput. Phys., 173:620–635,2001.Google Scholar
[34]Oevermann, M., Scharfenberg, C., and Klein, R.. A sharp interface finite volume method for elliptic equations on cartesian grids. J. Comput. Phys., 228:5184–5206,2009.CrossRefGoogle Scholar
[35]Osher, S. and Fedkiw, R.. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2003.Google Scholar
[36]Osher, S. and Sethian, J. A.. Fronts propagating with curvature-dependent speed: Algorithms based on hamiltonVjacobi formulations. J. Comput. Phys., 79(12), 1988.Google Scholar
[37]Saad, Y.. Sparskit a basic tool-kit for sparse matrix computations. http://www-users.cs.umn.edu/~saad/software/SPARSKIT/sparskit.html.Google Scholar
[38]Sarthou, A., Vincent, S., Angot, P., and Caltagirone, J.P.. The algebraic immersed interface and boundary method for elliptic equations with discontinuous coefficients. submitted, 2009.Google Scholar
[39]Sethian, J. A.. Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge, UK, 1999.Google Scholar
[40]Sethian, J. A.. Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys., 169:503–555,2001.Google Scholar
[41]Svard, M. and Nordstrom, J.. On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys., 218:333–352,2006.Google Scholar
[42]Wiegmann, A. and Bube, K.. The explicit jump immersed interface method: finite difference method for pdes with piecewise smooth solutions. SIAM J. Numer. Anal., 37(3):827–862, 2000.Google Scholar
[43]Yu, S. a and Wei, G.W.. Three-dimensional matched interface and boundary (mib) method for treating geometric singularities. J. Comput. Phys., 227:602–632,2007.Google Scholar
[44]Zhong, X.. A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity. J. Comput. Phys., 225:1066–1099,2007.Google Scholar
[45]Zhou, Y. C. and Wei, G. W.. On the fictitious-domain and interpolation formulations of the matched interface and boundary (mib) method. J. Comput. Phys., 219:228–246,2006.Google Scholar
[46]Zhou, Y. C., Zhao, S., Feig, M., and Wei, G. W.. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys., 213:1–30,2006.Google Scholar