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Volumn 7, Issue 1, 2009, Pages 81-107

Solving PDES in complex geometries: A diffuse domain approach

Author keywords

Adaptive finite difference methods; Adaptive finite element methods; Complex geometry; Diffuse interface; Multigrid methods; Partial differential equations; Phase field approximation

Indexed keywords


EID: 64549103448     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2009.v7.n1.a4     Document Type: Article
Times cited : (215)

References (49)
  • 1
    • 0002108952 scopus 로고    scopus 로고
    • The fast construction of extension velocities in level set methods
    • D. Adalsteinsson and J. Sethian, The fast construction of extension velocities in level set methods, J. Compt. Phys., 148, 2-22, 1999.
    • (1999) J. Compt. Phys , vol.148 , pp. 2-22
    • Adalsteinsson, D.1    Sethian, J.2
  • 2
    • 2342502809 scopus 로고    scopus 로고
    • Finite element method for epitaxial growth with attachment-detachment kinetics
    • E. Bäensch, F. Haußer, O. Lakkis, B. Li and A. Voigt, Finite element method for epitaxial growth with attachment-detachment kinetics, J. Comput. Phys., 194, 409-434, 2004.
    • (2004) J. Comput. Phys , vol.194 , pp. 409-434
    • Bäensch, E.1    Haußer, F.2    Lakkis, O.3    Li, B.4    Voigt, A.5
  • 3
    • 33845723210 scopus 로고    scopus 로고
    • Spectral methods for partial differential equations in irregular domains: The spectral smoothed boundary method
    • A. Bueno-Orovio, V.M. Perez-Garcia and F.H. Fenton, Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method, SIAM J. Sci. Comput., 28, 886-900, 2006.
    • (2006) SIAM J. Sci. Comput , vol.28 , pp. 886-900
    • Bueno-Orovio, A.1    Perez-Garcia, V.M.2    Fenton, F.H.3
  • 4
    • 33645013922 scopus 로고    scopus 로고
    • Spectral smoothed boundary methods: The role of external boundary conditions
    • A. Bueno-Orovio and V.M. Perez-Garcia, Spectral smoothed boundary methods: the role of external boundary conditions, Numer. Meth. Partial Diff. Eqns., 22, 435-448, 2006.
    • (2006) Numer. Meth. Partial Diff. Eqns , vol.22 , pp. 435-448
    • Bueno-Orovio, A.1    Perez-Garcia, V.M.2
  • 5
    • 60349116553 scopus 로고    scopus 로고
    • Finite element approximation of elliptic partial differential equations on implicit surfaces
    • M. Burger, Finite element approximation of elliptic partial differential equations on implicit surfaces, Comp. Vis. Sci., 12, 87-100, 2009.
    • (2009) Comp. Vis. Sci , vol.12 , pp. 87-100
    • Burger, M.1
  • 6
    • 0001314971 scopus 로고
    • Dynamics of layered interfaces arising from phase boundaries
    • G. Caginalp and P.C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48, 506-518, 1988.
    • (1988) SIAM J. Appl. Math , vol.48 , pp. 506-518
    • Caginalp, G.1    Fife, P.C.2
  • 7
    • 17544389134 scopus 로고    scopus 로고
    • Generalized finite element methods for three-dimensional structural mechanics problems
    • C.A. Duarte, I. Babuska and J.T. Oden, Generalized finite element methods for three-dimensional structural mechanics problems, Comp. Struct., 77, 215-232, 2000.
    • (2000) Comp. Struct , vol.77 , pp. 215-232
    • Duarte, C.A.1    Babuska, I.2    Oden, J.T.3
  • 8
    • 42549098645 scopus 로고    scopus 로고
    • Advances of and by phase-field modelling in condensed-matter physics
    • H. Emmerich, Advances of and by phase-field modelling in condensed-matter physics, Adv. Phys., 57, 1-87, 2008.
    • (2008) Adv. Phys , vol.57 , pp. 1-87
    • Emmerich, H.1
  • 9
    • 0346837922 scopus 로고    scopus 로고
    • A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
    • R.P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Compt. Phys., 152, 457-492, 1999.
    • (1999) J. Compt. Phys , vol.152 , pp. 457-492
    • Fedkiw, R.P.1    Aslam, T.2    Merriman, B.3    Osher, S.4
  • 10
    • 17744397787 scopus 로고    scopus 로고
    • Modeling wave propagation in realistic heart geometries using the phase-field method
    • F.H. Fenton, E.M. Cherry, A. Karma and W.J. Rappel, Modeling wave propagation in realistic heart geometries using the phase-field method, Chaos, 15, 013502, 2005.
    • (2005) Chaos , vol.15 , pp. 013502
    • Fenton, F.H.1    Cherry, E.M.2    Karma, A.3    Rappel, W.J.4
  • 11
    • 0003016830 scopus 로고
    • Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter
    • P.C. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter, Elect. J. Diff. Eqs., 16, 1-49, 1995.
    • (1995) Elect. J. Diff. Eqs , vol.16 , pp. 1-49
    • Fife, P.C.1    Penrose, O.2
  • 12
    • 10244220985 scopus 로고    scopus 로고
    • A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem
    • F. Gibou and R. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Compt. Phys., 202, 577-601, 2005.
    • (2005) J. Compt. Phys , vol.202 , pp. 577-601
    • Gibou, F.1    Fedkiw, R.2
  • 13
    • 0037050814 scopus 로고    scopus 로고
    • A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
    • F. Gibou, R. Fedkiw, L.T. Cheng and M. Kang, A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Compt. Phys., 176, 205-227, 2002.
    • (2002) J. Compt. Phys , vol.176 , pp. 205-227
    • Gibou, F.1    Fedkiw, R.2    Cheng, L.T.3    Kang, M.4
  • 14
    • 0019392895 scopus 로고
    • A numerical method for two phase flow with an unstable interface
    • J. Glimm, D. Marchesin and O. McBryan, A numerical method for two phase flow with an unstable interface, J. Compt. Phys., 39, 179-200, 1981.
    • (1981) J. Compt. Phys , vol.39 , pp. 179-200
    • Glimm, J.1    Marchesin, D.2    McBryan, O.3
  • 15
    • 0028035577 scopus 로고
    • A fictitious domain method for external incompressible viscous-flow modeled by Navier-Stokes equations
    • R. Glowinski, T.W. Pan and J. Periaux, A fictitious domain method for external incompressible viscous-flow modeled by Navier-Stokes equations, Comput. Meth. Appl. Mech. Engin., 112, 133-148, 1994.
    • (1994) Comput. Meth. Appl. Mech. Engin , vol.112 , pp. 133-148
    • Glowinski, R.1    Pan, T.W.2    Periaux, J.3
  • 16
    • 0010207466 scopus 로고    scopus 로고
    • Wavelet and finite element solutions for the Neumann problem using fictitious domains
    • R. Glowinski, T.W. Pan, R.O. Wells and X.D. Zhou, Wavelet and finite element solutions for the Neumann problem using fictitious domains, J. Comput. Phys., 126, 40-51, 1996.
    • (1996) J. Comput. Phys , vol.126 , pp. 40-51
    • Glowinski, R.1    Pan, T.W.2    Wells, R.O.3    Zhou, X.D.4
  • 17
    • 55549138751 scopus 로고    scopus 로고
    • Immersed-interface finite-element method for elliptic interface problems with non-homogeneous jump conditions
    • Y. Gong, B. Li and Z. Li, Immersed-interface finite-element method for elliptic interface problems with non-homogeneous jump conditions, SIAM J. Numer. Anal., 46, 472-495, 2008.
    • (2008) SIAM J. Numer. Anal , vol.46 , pp. 472-495
    • Gong, Y.1    Li, B.2    Li, Z.3
  • 18
    • 0031538248 scopus 로고    scopus 로고
    • Composite finite elements for the approximation of PDEs on domains with complicated micro-structures
    • W. Hackbusch and S. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Num. Math., 75, 447-472, 1997.
    • (1997) Num. Math , vol.75 , pp. 447-472
    • Hackbusch, W.1    Sauter, S.2
  • 19
    • 41649101086 scopus 로고    scopus 로고
    • F. Haußer, M.E. Jabbour and A. Voigt, A step-flow model for the heteroepitaxial growth of strained, substitutional, binary alloy films with phase segregation: I. theory, Multisc. Mod. Sim., 6, 158-189, 2007.
    • F. Haußer, M.E. Jabbour and A. Voigt, A step-flow model for the heteroepitaxial growth of strained, substitutional, binary alloy films with phase segregation: I. theory, Multisc. Mod. Sim., 6, 158-189, 2007.
  • 20
    • 33750329359 scopus 로고    scopus 로고
    • An efficient second-order accurate cut-cell method for solving the variable coefficient Poisson equation with jump conditions on irregular domains
    • H. Ji, F.S. Lien and E. Yee, An efficient second-order accurate cut-cell method for solving the variable coefficient Poisson equation with jump conditions on irregular domains, Int. J. Num. Meth. Fluids, 52, 723-748, 2006.
    • (2006) Int. J. Num. Meth. Fluids , vol.52 , pp. 723-748
    • Ji, H.1    Lien, F.S.2    Yee, E.3
  • 21
    • 0001827481 scopus 로고    scopus 로고
    • A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
    • P. Colella, D. Graves, T. Ligocki, D. Trebotich and B. Ustraalen A Cartesian grid embedded boundary method for Poisson's equation on irregular domains, J. Compt. Phys., 147, 60-85, 1998.
    • (1998) J. Compt. Phys , vol.147 , pp. 60-85
    • Colella, P.1    Graves, D.2    Ligocki, T.3    Trebotich, D.4    Ustraalen, B.5
  • 22
    • 65649144169 scopus 로고    scopus 로고
    • Embedded boundary algorithms and software for partial differential equations
    • H. Johansen and P. Colella, Embedded boundary algorithms and software for partial differential equations, J. Phys., 125, 012084, 2008.
    • (2008) J. Phys , vol.125 , pp. 012084
    • Johansen, H.1    Colella, P.2
  • 23
    • 37649027519 scopus 로고    scopus 로고
    • Computational approach for modeling intra- and extracellular dynamics
    • J. Kockelkoren, H. Levine and W.-J. Rappel, Computational approach for modeling intra- and extracellular dynamics, Phys. Rev. E, 68, 037702, 2003.
    • (2003) Phys. Rev. E , vol.68 , pp. 037702
    • Kockelkoren, J.1    Levine, H.2    Rappel, W.-J.3
  • 24
    • 0028485385 scopus 로고
    • The immersed interface method for elliptic-equations with discontinuous coefficients and singular sources
    • R.J. LeVeque and Z.L. Li, The immersed interface method for elliptic-equations with discontinuous coefficients and singular sources, SIAM J. Num. Anal., 31, 1019-1044, 1994.
    • (1994) SIAM J. Num. Anal , vol.31 , pp. 1019-1044
    • LeVeque, R.J.1    Li, Z.L.2
  • 25
    • 33244490736 scopus 로고    scopus 로고
    • Membrane-bound Turing patterns
    • H. Levine and W.J. Rappel, Membrane-bound Turing patterns, Phys. Rev. E, 72, 061912, 2005.
    • (2005) Phys. Rev. E , vol.72 , pp. 061912
    • Levine, H.1    Rappel, W.J.2
  • 26
    • 35848943928 scopus 로고    scopus 로고
    • The immersed interface method: Numerical solutions of PDEs involving interfaces and irregular domains
    • Z. Li and K. Ito, The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains, SIAM Front. Appl. Math., 33, 2006.
    • (2006) SIAM Front. Appl. Math , vol.33
    • Li, Z.1    Ito, K.2
  • 27
    • 0742271742 scopus 로고    scopus 로고
    • New cartesian grid methods for interface problems using the finite element formulation
    • Z.L. Li, T. Lin and X.H. Wu, New cartesian grid methods for interface problems using the finite element formulation, Num. Math., 96, 61-98, 2003.
    • (2003) Num. Math , vol.96 , pp. 61-98
    • Li, Z.L.1    Lin, T.2    Wu, X.H.3
  • 29
    • 65449146384 scopus 로고    scopus 로고
    • Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding and fission
    • to appear
    • J. Lowengrub, A. Rätz and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding and fission, Phys. Rev. E, to appear.
    • Phys. Rev. E
    • Lowengrub, J.1    Rätz, A.2    Voigt, A.3
  • 30
    • 10844239997 scopus 로고    scopus 로고
    • Evolving interfaces via gradients of geometry-dependent interior Poisson problems: Application to tumor growth
    • P. Macklin and J. Lowengrub, Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth, J. Compt. Phys., 203, 191-220, 2005.
    • (2005) J. Compt. Phys , vol.203 , pp. 191-220
    • Macklin, P.1    Lowengrub, J.2
  • 31
    • 49749145984 scopus 로고    scopus 로고
    • A new ghost cell/level set method for moving boundary problems: Application to tumor growth
    • P. Macklin and J. Lowengrub, A new ghost cell/level set method for moving boundary problems: application to tumor growth, J. Sci. Comput., 35, 266-299, 2008.
    • (2008) J. Sci. Comput , vol.35 , pp. 266-299
    • Macklin, P.1    Lowengrub, J.2
  • 32
    • 23044476665 scopus 로고    scopus 로고
    • Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations
    • S. Marella, S. Krishnan, H. Liu and H.S. Udaykumar, Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations, J. Compt. Phys., 210, 1-31, 2005.
    • (2005) J. Compt. Phys , vol.210 , pp. 1-31
    • Marella, S.1    Krishnan, S.2    Liu, H.3    Udaykumar, H.S.4
  • 33
    • 0035505590 scopus 로고    scopus 로고
    • A Cartesian grid embedded boundary method for the heat equation on irregular domains
    • P. McCorquodale, P. Colella and H. Johansen, A Cartesian grid embedded boundary method for the heat equation on irregular domains, J. Compt. Phys., 173, 620-635, 2001.
    • (2001) J. Compt. Phys , vol.173 , pp. 620-635
    • McCorquodale, P.1    Colella, P.2    Johansen, H.3
  • 34
    • 0030380578 scopus 로고    scopus 로고
    • The partition of unity finite element method: Basic theory and applications
    • J.M. Melenk and I. Babuska, The partition of unity finite element method: basic theory and applications, Comp. Meth. Appl. Mech. Eng., 139, 289-314, 1996.
    • (1996) Comp. Meth. Appl. Mech. Eng , vol.139 , pp. 289-314
    • Melenk, J.M.1    Babuska, I.2
  • 35
    • 34548697081 scopus 로고    scopus 로고
    • Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology
    • I.L. Novak, F. Gao, Y.S. Choi, D. Resasco, J.C. Schaff and B.M. Slepchenko, Diffusion on a curved surface coupled to diffusion in the volume: application to cell biology, J. Compt. Phys., 226, 1271-1290, 2007.
    • (2007) J. Compt. Phys , vol.226 , pp. 1271-1290
    • Novak, I.L.1    Gao, F.2    Choi, Y.S.3    Resasco, D.4    Schaff, J.C.5    Slepchenko, B.M.6
  • 37
    • 0001410452 scopus 로고
    • Front migration in the nonlinear Cahn-Hilliard equation
    • R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London A, 422, 261-278, 1989.
    • (1989) Proc. Roy. Soc. London A , vol.422 , pp. 261-278
    • Pego, R.1
  • 39
    • 34547574651 scopus 로고    scopus 로고
    • A general fictitious domain method with immersed jumps and multilevel nested structured meshes
    • I. Ramiere, P. Angot and M. Belliard, A general fictitious domain method with immersed jumps and multilevel nested structured meshes, J. Compt. Phys., 225, 1347-1387, 2007.
    • (2007) J. Compt. Phys , vol.225 , pp. 1347-1387
    • Ramiere, I.1    Angot, P.2    Belliard, M.3
  • 40
    • 34548678944 scopus 로고    scopus 로고
    • PDE's on surfaces - a diffuse interface approach
    • A. Rätz and A. Voigt, PDE's on surfaces - a diffuse interface approach, Commun. Math. Sci., 4, 575-590, 2006.
    • (2006) Commun. Math. Sci , vol.4 , pp. 575-590
    • Rätz, A.1    Voigt, A.2
  • 41
    • 31744451387 scopus 로고    scopus 로고
    • Two-scale composite finite element method for Dirichlet problems on complicated domains
    • M. Rech, S. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Num. Math., 102, 681-708, 2006.
    • (2006) Num. Math , vol.102 , pp. 681-708
    • Rech, M.1    Sauter, S.2    Smolianski, A.3
  • 42
    • 64549163624 scopus 로고    scopus 로고
    • A nonconforming extended finite element method for interface problems
    • submitted
    • A. Ribalta and A. Voigt, A nonconforming extended finite element method for interface problems, SIAM J. Num. Anal., submitted.
    • SIAM J. Num. Anal
    • Ribalta, A.1    Voigt, A.2
  • 43
    • 26944447097 scopus 로고    scopus 로고
    • A Cartesian grid embedded boundary method for the heat equation and Poisson's equation in three dimensions
    • P. Schwartz, M. Barad, P. Colella and T. Ligocki, A Cartesian grid embedded boundary method for the heat equation and Poisson's equation in three dimensions, J. Compt. Phys., 211, 531-550, 2006.
    • (2006) J. Compt. Phys , vol.211 , pp. 531-550
    • Schwartz, P.1    Barad, M.2    Colella, P.3    Ligocki, T.4
  • 44
    • 44149097105 scopus 로고    scopus 로고
    • Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing
    • J.A. Sethian and Y. Shan, Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing, J. Compt. Phys., 227, 6411-6447, 2008.
    • (2008) J. Compt. Phys , vol.227 , pp. 6411-6447
    • Sethian, J.A.1    Shan, Y.2
  • 45
    • 20344390525 scopus 로고    scopus 로고
    • AMDiS - adaptive multidimensional simulations: Composite finite elements and signed distance functions
    • C. Stöcker, S. Vey and A. Voigt, AMDiS - adaptive multidimensional simulations: composite finite elements and signed distance functions, WSEAC Trans. Circuits and Systems, 4, 111-116, 2005.
    • (2005) WSEAC Trans. Circuits and Systems , vol.4 , pp. 111-116
    • Stöcker, C.1    Vey, S.2    Voigt, A.3
  • 46
    • 0033279022 scopus 로고    scopus 로고
    • An efficient, interface preserving level-set redistancing algorithm and its application to interfacial incompressible fluid flow
    • M. Sussman and E. Fatemi, An efficient, interface preserving level-set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. Sci. Comput., 20, 1165-1191, 1999.
    • (1999) SIAM J. Sci. Comput , vol.20 , pp. 1165-1191
    • Sussman, M.1    Fatemi, E.2
  • 47
    • 33847313675 scopus 로고    scopus 로고
    • AMDiS - adaptive multidimensional simulations
    • S. Vey and A. Voigt, AMDiS - adaptive multidimensional simulations, Comput. Visual. Sci., 10, 57-67, 2007.
    • (2007) Comput. Visual. Sci , vol.10 , pp. 57-67
    • Vey, S.1    Voigt, A.2
  • 48
    • 34548460677 scopus 로고    scopus 로고
    • Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method
    • S.M. Wise, J.S. Kim and J. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Compt. Phys., 226, 414-446, 2007.
    • (2007) J. Compt. Phys , vol.226 , pp. 414-446
    • Wise, S.M.1    Kim, J.S.2    Lowengrub, J.3
  • 49
    • 29144444774 scopus 로고    scopus 로고
    • A level-set method for interfacial flows with surfactant
    • J.J. Xu, Z. Li, J. Lowengrub and H. Zhao, A level-set method for interfacial flows with surfactant, J. Compt. Phys., 212, 590-616, 2006.
    • (2006) J. Compt. Phys , vol.212 , pp. 590-616
    • Xu, J.J.1    Li, Z.2    Lowengrub, J.3    Zhao, H.4


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