메뉴 건너뛰기




Volumn 227, Issue 1, 2007, Pages 472-491

Local discontinuous Galerkin methods for the Cahn-Hilliard type equations

Author keywords

Cahn Hilliard equation; Cahn Hilliard system; Local discontinuous Galerkin methods; Stability

Indexed keywords

GALERKIN METHODS; NONLINEAR EQUATIONS;

EID: 35348877949     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.08.001     Document Type: Article
Times cited : (133)

References (37)
  • 1
    • 0036410391 scopus 로고    scopus 로고
    • Unified analysis of discontinuous Galerkin methods for elliptic problems
    • Arnold D.N., Brezzi F., Cockburn B., and Marini L.D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779
    • (2002) SIAM J. Numer. Anal. , vol.39 , pp. 1749-1779
    • Arnold, D.N.1    Brezzi, F.2    Cockburn, B.3    Marini, L.D.4
  • 2
    • 0031500465 scopus 로고    scopus 로고
    • Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy
    • Barrett J.W., and Blowey J.F. Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77 (1997) 1-34
    • (1997) Numer. Math. , vol.77 , pp. 1-34
    • Barrett, J.W.1    Blowey, J.F.2
  • 3
    • 0007425693 scopus 로고    scopus 로고
    • Finite element approximation of the Cahn-Hilliard equation with degenerate mobility
    • Barrett J.W., Blowey J.F., and Garcke H. Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286-318
    • (1999) SIAM J. Numer. Anal. , vol.37 , pp. 286-318
    • Barrett, J.W.1    Blowey, J.F.2    Garcke, H.3
  • 4
    • 0035537706 scopus 로고    scopus 로고
    • On fully practical finite element approximations of degenerate Cahn-Hilliard systems
    • Barrett J.W., Blowey J.F., and Garcke H. On fully practical finite element approximations of degenerate Cahn-Hilliard systems. M2AN Math. Model. Numer. Anal. 35 (2001) 713-748
    • (2001) M2AN Math. Model. Numer. Anal. , vol.35 , pp. 713-748
    • Barrett, J.W.1    Blowey, J.F.2    Garcke, H.3
  • 5
    • 0001416032 scopus 로고    scopus 로고
    • A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations
    • Bassi F., and Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267-279
    • (1997) J. Comput. Phys. , vol.131 , pp. 267-279
    • Bassi, F.1    Rebay, S.2
  • 6
    • 84972038069 scopus 로고
    • The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis
    • Blowey J.F., and Elliott C.M. The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3 (1992) 147-179
    • (1992) Eur. J. Appl. Math. , vol.3 , pp. 147-179
    • Blowey, J.F.1    Elliott, C.M.2
  • 7
    • 0030515299 scopus 로고    scopus 로고
    • Numerical analysis of a model for phase separation of multi-component alloy
    • Blowey J.F., Copetti M.I.M., and Elliott C.M. Numerical analysis of a model for phase separation of multi-component alloy. IMA J. Numer. Anal. 16 (1996) 111-139
    • (1996) IMA J. Numer. Anal. , vol.16 , pp. 111-139
    • Blowey, J.F.1    Copetti, M.I.M.2    Elliott, C.M.3
  • 8
    • 33746012315 scopus 로고
    • Free energy of non-uniform system. I. Interfacial free energy
    • Cahn J.W., and Hilliard J.E. Free energy of non-uniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258-267
    • (1958) J. Chem. Phys. , vol.28 , pp. 258-267
    • Cahn, J.W.1    Hilliard, J.E.2
  • 9
    • 19944391320 scopus 로고    scopus 로고
    • A discontinuous Galerkin method for the Cahn-Hilliard equation
    • Choo S.M., and Lee Y.J. A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Appl. Math. Comput. 18 (2005) 113-126
    • (2005) J. Appl. Math. Comput. , vol.18 , pp. 113-126
    • Choo, S.M.1    Lee, Y.J.2
  • 10
    • 84966261380 scopus 로고
    • The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case
    • Cockburn B., Hou S., and Shu C.-W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case. Math. Comp. 54 (1990) 545-581
    • (1990) Math. Comp. , vol.54 , pp. 545-581
    • Cockburn, B.1    Hou, S.2    Shu, C.-W.3
  • 11
    • 45249130080 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: One-dimensional systems
    • Cockburn B., Lin S.-Y., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: One-dimensional systems. J. Comput. Phys. 84 (1989) 90-113
    • (1989) J. Comput. Phys. , vol.84 , pp. 90-113
    • Cockburn, B.1    Lin, S.-Y.2    Shu, C.-W.3
  • 12
    • 84966246413 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: General framework
    • Cockburn B., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: General framework. Math. Comp. 52 (1989) 411-435
    • (1989) Math. Comp. , vol.52 , pp. 411-435
    • Cockburn, B.1    Shu, C.-W.2
  • 13
    • 0001690553 scopus 로고    scopus 로고
    • The Runge-Kutta discontinuous Galerkin method for conservation laws. V: Multidimensional systems
    • Cockburn B., and Shu C.-W. The Runge-Kutta discontinuous Galerkin method for conservation laws. V: Multidimensional systems. J. Comput. Phys. 141 (1998) 199-224
    • (1998) J. Comput. Phys. , vol.141 , pp. 199-224
    • Cockburn, B.1    Shu, C.-W.2
  • 14
    • 0344393366 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for time-dependent convection-diffusion systems
    • Cockburn B., and Shu C.-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 2440-2463
    • Cockburn, B.1    Shu, C.-W.2
  • 15
    • 21144473934 scopus 로고
    • Numerical analysis of the Cahn-Hilliard equation with logarithmic free energy
    • Copetti M.I.M., and Elliott C.M. Numerical analysis of the Cahn-Hilliard equation with logarithmic free energy. Numer. Math. 63 (1992) 39-65
    • (1992) Numer. Math. , vol.63 , pp. 39-65
    • Copetti, M.I.M.1    Elliott, C.M.2
  • 16
    • 0000662072 scopus 로고
    • A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation
    • Elliott C.M., and French D.A. A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26 (1989) 884-903
    • (1989) SIAM J. Numer. Anal. , vol.26 , pp. 884-903
    • Elliott, C.M.1    French, D.A.2
  • 17
    • 0001475208 scopus 로고
    • A second order splitting method for the Cahn-Hilliard equation
    • Elliott C.M., French D.A., and Milner F.A. A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54 (1989) 575-590
    • (1989) Numer. Math. , vol.54 , pp. 575-590
    • Elliott, C.M.1    French, D.A.2    Milner, F.A.3
  • 18
    • 0030526461 scopus 로고    scopus 로고
    • On the Cahn-Hilliard equation with degenerate mobility
    • Elliott C.M., and Garcke H. On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996) 404-423
    • (1996) SIAM J. Math. Anal. , vol.27 , pp. 404-423
    • Elliott, C.M.1    Garcke, H.2
  • 19
    • 0027839918 scopus 로고
    • Systems of Cahn-Hilliard equations
    • Eyre D.J. Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686-1712
    • (1993) SIAM J. Appl. Math. , vol.53 , pp. 1686-1712
    • Eyre, D.J.1
  • 20
    • 1642633953 scopus 로고    scopus 로고
    • Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits
    • Feng X., and Prohl A. Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comp. 73 (2004) 541-567
    • (2004) Math. Comp. , vol.73 , pp. 541-567
    • Feng, X.1    Prohl, A.2
  • 21
    • 35348833857 scopus 로고    scopus 로고
    • Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition
    • Feng X.B., and Karakashian O.A. Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition. Math. Comp. 76 (2007) 1093-1117
    • (2007) Math. Comp. , vol.76 , pp. 1093-1117
    • Feng, X.B.1    Karakashian, O.A.2
  • 22
    • 0041079077 scopus 로고    scopus 로고
    • A stable and conservative finite difference scheme for the Cahn-Hilliard equation
    • Furihata D. A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87 (2001) 675-699
    • (2001) Numer. Math. , vol.87 , pp. 675-699
    • Furihata, D.1
  • 23
    • 29144470271 scopus 로고    scopus 로고
    • A multigrid finite element solver for the Cahn-Hilliard equation
    • Kay D., and Welford R. A multigrid finite element solver for the Cahn-Hilliard equation. J. Comput. Phys. 212 (2006) 288-304
    • (2006) J. Comput. Phys. , vol.212 , pp. 288-304
    • Kay, D.1    Welford, R.2
  • 24
    • 0346332035 scopus 로고    scopus 로고
    • Conservative multigrid methods for Cahn-Hilliard fluids
    • Kim J., Kang K., and Lowengrub J. Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511-543
    • (2004) J. Comput. Phys. , vol.193 , pp. 511-543
    • Kim, J.1    Kang, K.2    Lowengrub, J.3
  • 25
    • 85128845199 scopus 로고    scopus 로고
    • Conservative multigrid methods for ternary Cahn-Hilliard systems
    • Kim J., Kang K., and Lowengrub J. Conservative multigrid methods for ternary Cahn-Hilliard systems. Commun. Math. Sci. 2 (2004) 53-77
    • (2004) Commun. Math. Sci. , vol.2 , pp. 53-77
    • Kim, J.1    Kang, K.2    Lowengrub, J.3
  • 26
    • 2442475431 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for nonlinear dispersive equations
    • Levy D., Shu C.-W., and Yan J. Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196 (2004) 751-772
    • (2004) J. Comput. Phys. , vol.196 , pp. 751-772
    • Levy, D.1    Shu, C.-W.2    Yan, J.3
  • 27
    • 0345875566 scopus 로고
    • Spinodal decomposition in ternary systems
    • Morral J.E., and Cahn J.W. Spinodal decomposition in ternary systems. Acta Metall. 19 (1971) 1037-1045
    • (1971) Acta Metall. , vol.19 , pp. 1037-1045
    • Morral, J.E.1    Cahn, J.W.2
  • 28
    • 85189851476 scopus 로고    scopus 로고
    • W.H. Reed, T.R. Hill, Triangular mesh method for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.
  • 29
    • 45449125925 scopus 로고
    • Efficient implementation of essentially non-oscillatory shock-capturing schemes
    • Shu C.-W., and Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439-471
    • (1988) J. Comput. Phys. , vol.77 , pp. 439-471
    • Shu, C.-W.1    Osher, S.2
  • 30
    • 21844485652 scopus 로고
    • A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation
    • Sun Z.Z. A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation. Math. Comp. 64 (1995) 1463-1471
    • (1995) Math. Comp. , vol.64 , pp. 1463-1471
    • Sun, Z.Z.1
  • 31
    • 33748907906 scopus 로고    scopus 로고
    • A discontinuous Galerkin method for the Cahn-Hilliard equation
    • Wells G.N., Kuhl E., and Garikipati K. A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys. 218 (2006) 860-877
    • (2006) J. Comput. Phys. , vol.218 , pp. 860-877
    • Wells, G.N.1    Kuhl, E.2    Garikipati, K.3
  • 32
    • 3042791844 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for three classes of nonlinear wave equations
    • Xu Y., and Shu C.-W. Local discontinuous Galerkin methods for three classes of nonlinear wave equations. J. Comput. Math. 22 (2004) 250-274
    • (2004) J. Comput. Math. , vol.22 , pp. 250-274
    • Xu, Y.1    Shu, C.-W.2
  • 33
    • 16844366209 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for nonlinear Schrödinger equations
    • Xu Y., and Shu C.-W. Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205 (2005) 72-97
    • (2005) J. Comput. Phys. , vol.205 , pp. 72-97
    • Xu, Y.1    Shu, C.-W.2
  • 34
    • 22544467513 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations
    • Xu Y., and Shu C.-W. Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations. Physica D 208 (2005) 21-58
    • (2005) Physica D , vol.208 , pp. 21-58
    • Xu, Y.1    Shu, C.-W.2
  • 35
    • 33645906589 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations
    • Xu Y., and Shu C.-W. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Comput. Method Appl. Mech. Eng. 195 (2006) 3430-3447
    • (2006) Comput. Method Appl. Mech. Eng. , vol.195 , pp. 3430-3447
    • Xu, Y.1    Shu, C.-W.2
  • 36
    • 0038693389 scopus 로고    scopus 로고
    • A local discontinuous Galerkin method for KdV type equations
    • Yan J., and Shu C.-W. A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769-791
    • (2002) SIAM J. Numer. Anal. , vol.40 , pp. 769-791
    • Yan, J.1    Shu, C.-W.2
  • 37
    • 1942475854 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for partial differential equations with higher order derivatives
    • Yan J., and Shu C.-W. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comp. 17 (2002) 27-47
    • (2002) J. Sci. Comp. , vol.17 , pp. 27-47
    • Yan, J.1    Shu, C.-W.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.