메뉴 건너뛰기




Volumn 27, Issue 1, 2011, Pages 160-196

Goal-oriented error estimation for Cahn-Hilliard models of binary phase transition

Author keywords

a posteriori error estimation; Cahn Hilliard; diffuse interface; dual Cahn Hilliard; dual well posedness; error residual equivalence; goal oriented error analysis; linearized adjoint; quantities of interest

Indexed keywords

NONLINEAR EQUATIONS;

EID: 78649884402     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20638     Document Type: Article
Times cited : (32)

References (75)
  • 1
    • 33746012315 scopus 로고
    • Free energy of a nonuniform system. I. Interfacial free energy
    • J. W. Cahn, and, J. E. Hilliard, Free energy of a nonuniform 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
  • 3
    • 0041152882 scopus 로고    scopus 로고
    • Quasi-incompressible Cahn-Hilliard fluids and topological transitions
    • J. S. Lowengrub, and, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc R Soc Lond Ser A Math Phys Eng Sci 454 (1998), 2617-2654.
    • (1998) Proc R Soc Lond ser A Math Phys Eng Sci , vol.454 , pp. 2617-2654
    • Lowengrub, J.S.1    Truskinovsky, L.2
  • 5
    • 0036036465 scopus 로고    scopus 로고
    • Phase-field models for microstructure evolution
    • L.-Q. Chen, Phase-field models for microstructure evolution, Annu Rev Mater Res 32 (2002), 113-140.
    • (2002) Annu Rev Mater Res , vol.32 , pp. 113-140
    • Chen, L.-Q.1
  • 6
    • 77951611471 scopus 로고    scopus 로고
    • General diffuse-interface theories and an approach to predictive tumor growth modeling
    • J. Oden, A. Hawkins, and, S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math Models Methods Appl Sci 20 (2010), 477-517.
    • (2010) Math Models Methods Appl Sci , vol.20 , pp. 477-517
    • Oden, J.1    Hawkins, A.2    Prudhomme, S.3
  • 7
    • 0022862318 scopus 로고
    • On the Cahn-Hilliard equation
    • C. Elliott, and, S. Zheng, On the Cahn-Hilliard equation, Arch Ration Mech Anal 96 (1986), 339-357.
    • (1986) Arch Ration Mech Anal , vol.96 , pp. 339-357
    • Elliott, C.1    Zheng, S.2
  • 9
    • 84971972596 scopus 로고
    • The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis
    • J. Blowey, and, C. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis, European J Appl Math 2 (1991), 233-279.
    • (1991) European J Appl Math , vol.2 , pp. 233-279
    • Blowey, J.1    Elliott, C.2
  • 10
    • 0007373998 scopus 로고
    • A generalized equation for phase separation of a multi-component mixture with interfacial free energy
    • University of Minnesota, Minneapolis, Minnesota.
    • C. M. Elliott, and, S. Luckhaus, A generalized equation for phase separation of a multi-component mixture with interfacial free energy, IMA Preprint Series 887, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota, 1991.
    • (1991) IMA Preprint Series 887, Institute for Mathematics and Its Applications
    • Elliott, C.M.1    Luckhaus, S.2
  • 11
    • 0030526461 scopus 로고    scopus 로고
    • On the Cahn-Hilliard equation with degenerate mobility
    • C. Elliott, and, H. Garcke, 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.1    Garcke, H.2
  • 12
    • 0033450285 scopus 로고    scopus 로고
    • Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility
    • J. Barrett, and, J. Blowey, Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility, Math Comp 68 (1999), 487-517.
    • (1999) Math Comp , vol.68 , pp. 487-517
    • Barrett, J.1    Blowey, J.2
  • 13
  • 14
  • 15
    • 21844492512 scopus 로고
    • Convergence of the Cahn-Hilliard equation to the Hele-Shaw model
    • N. D. Alikakos, P. W. Bates, and, X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch Ration Mech Anal 128 (1994), 165-205.
    • (1994) Arch Ration Mech Anal , vol.128 , pp. 165-205
    • Alikakos, N.D.1    Bates, P.W.2    Chen, X.3
  • 16
    • 0001514990 scopus 로고    scopus 로고
    • Global asymptotic limit of solutions of the Cahn-Hilliard equation
    • X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J Differential Geom 44 (1996), 262-311.
    • (1996) J Differential Geom , vol.44 , pp. 262-311
    • Chen, X.1
  • 17
    • 49349117199 scopus 로고    scopus 로고
    • The Willmore functional and instabilities in the Cahn-Hilliard equation
    • M. Burger, S.-Y. Chu, P. A. Markowich, and, C.-B. Schönlieb, The Willmore functional and instabilities in the Cahn-Hilliard equation, Commun Math Sci 6 (2008), 309-329.
    • (2008) Commun Math Sci , vol.6 , pp. 309-329
    • Burger, M.1    Chu, S.-Y.2    Markowich, P.A.3    Schönlieb, C.-B.4
  • 18
    • 0032321134 scopus 로고    scopus 로고
    • Unconditionally gradient stable time marching the Cahn-Hilliard equation
    • L.-Q. Chen, R. K. Kalia, and A. M. Stoneham, editors Materials Research Society Symposium Proceedings, Materials Research Society, Warrendale, PA
    • D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, Computational and mathematical models of microstructural evolution, J. W. Bullard, L.-Q. Chen, R. K. Kalia, and, A. M. Stoneham, editors, Vol. 529: Materials Research Society Symposium Proceedings, Materials Research Society, Warrendale, PA, 1998, pp. 39-46.
    • (1998) Computational and Mathematical Models of Microstructural Evolution , vol.529 , pp. 39-46
    • Eyre, D.1    Bullard, J.W.2
  • 19
    • 34147135231 scopus 로고    scopus 로고
    • On large time-stepping methods for the Cahn-Hilliard equation
    • Y. He, Y. Liu, and, T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer Math 57 (2007), 616-628.
    • (2007) Appl. Numer Math , vol.57 , pp. 616-628
    • He, Y.1    Liu, Y.2    Tang, T.3
  • 20
    • 67349192151 scopus 로고    scopus 로고
    • A class of stable spectral methods for the Cahn-Hilliard equation
    • L. He, and, Y. Liu, A class of stable spectral methods for the Cahn-Hilliard equation, J Comput Phys 228 (2009), 5101-5110.
    • (2009) J Comput Phys , vol.228 , pp. 5101-5110
    • He, L.1    Liu, Y.2
  • 21
    • 11144341340 scopus 로고    scopus 로고
    • Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Part I: Error analysis under minimum regularities
    • University of Minnesota, Minneapolis, Minnesota, Revised version published as [43].
    • X. Feng, and, A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Part I: Error analysis under minimum regularities, IMA Preprint Series 1798, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota, 2001 Revised version published as [43].
    • (2001) IMA Preprint Series 1798, Institute for Mathematics and Its Applications
    • Feng, X.1    Prohl, A.2
  • 22
    • 34547512592 scopus 로고    scopus 로고
    • Stability analysis of large time-stepping methods for epitaxial growth models
    • C. Xu, and, T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J Numer Anal 44 (2006), 1759-1779.
    • (2006) SIAM J Numer Anal , vol.44 , pp. 1759-1779
    • Xu, C.1    Tang, T.2
  • 23
    • 77953024541 scopus 로고    scopus 로고
    • An energy-stable and convergent finite-difference scheme for the phase field crystal equation
    • S. M. Wise, C. Wang, and, J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J Numer Anal 47 (2009), 2269-2288.
    • (2009) SIAM J Numer Anal , vol.47 , pp. 2269-2288
    • Wise, S.M.1    Wang, C.2    Lowengrub, J.S.3
  • 24
    • 0001617217 scopus 로고
    • Numerical studies of the Cahn-Hilliard equation for phase separation
    • C. Elliott, and, D. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J Appl Math 38 (1987), 97-128.
    • (1987) IMA J Appl Math , vol.38 , pp. 97-128
    • Elliott, C.1    French, D.2
  • 25
    • 21144473934 scopus 로고
    • Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy
    • M. Copetti, and, C. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer Math 63 (1992), 39-65.
    • (1992) Numer Math , vol.63 , pp. 39-65
    • Copetti, M.1    Elliott, C.2
  • 26
    • 21844484383 scopus 로고
    • An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy
    • J. Barrett, and, J. Blowey, An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy, Numer Math 72 (1995), 1-20.
    • (1995) Numer Math , vol.72 , pp. 1-20
    • Barrett, J.1    Blowey, J.2
  • 27
    • 11144340252 scopus 로고    scopus 로고
    • Error analysis of a mixed finite element method for the Cahn-Hilliard equation
    • X. Feng, and, A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer Math 99 (2004), 47-84.
    • (2004) Numer Math , vol.99 , pp. 47-84
    • Feng, X.1    Prohl, A.2
  • 28
    • 33748907906 scopus 로고    scopus 로고
    • A discontinuous Galerkin method for the Cahn-Hilliard equation
    • G. N. Wells, E. Kuhl, and, K. Garikipati, 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
  • 29
    • 35348833857 scopus 로고    scopus 로고
    • Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition
    • X. Feng, and, O. A. Karakashian, 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.1    Karakashian, O.A.2
  • 30
    • 85013995016 scopus 로고    scopus 로고
    • Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection
    • D. Kay, V. Styles, and, E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM J Numer Anal 47 (2009), 2660-2685.
    • (2009) SIAM J Numer Anal , vol.47 , pp. 2660-2685
    • Kay, D.1    Styles, V.2    Süli, E.3
  • 32
    • 57149116624 scopus 로고    scopus 로고
    • A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow
    • X. Feng, and, H.-J. Wu, A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow, J Comput Math 26 (2008), 767-796.
    • (2008) J Comput Math , vol.26 , pp. 767-796
    • Feng, X.1    Wu, H.-J.2
  • 33
    • 77955078669 scopus 로고    scopus 로고
    • A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations
    • S. Bartels, and, R. Müller, A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations, Interfaces Free Bound 12 (2010), 45-73.
    • (2010) Interfaces Free Bound , vol.12 , pp. 45-73
    • Bartels, S.1    Müller, R.2
  • 34
    • 44449107046 scopus 로고    scopus 로고
    • Adaptive finite element methods for Cahn-Hilliard equations
    • L. Baňas, and, R. Nürnberg, Adaptive finite element methods for Cahn-Hilliard equations, J Comput Appl Math 218 (2008), 2-11.
    • (2008) J Comput Appl Math , vol.218 , pp. 2-11
    • Baňas, L.1    Nürnberg, R.2
  • 35
    • 68949132872 scopus 로고    scopus 로고
    • A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy
    • L. Baňas, and, R. Nürnberg, A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy, M2AN Math Model Numer Anal 43 (2009), 1003-1026.
    • (2009) M2AN Math Model Numer Anal , vol.43 , pp. 1003-1026
    • Baňas, L.1    Nürnberg, R.2
  • 36
    • 1342283759 scopus 로고    scopus 로고
    • A posteriori error control for the Allen-Cahn problem: Circumventing Gronwall's inequality
    • D. Kessler, R. H. Nochetto, and, A. Schmidt, A posteriori error control for the Allen-Cahn problem: Circumventing Gronwall's inequality, M2AN Math Model Numer Anal 38 (2004), 129-142.
    • (2004) M2AN Math Model Numer Anal , vol.38 , pp. 129-142
    • Kessler, D.1    Nochetto, R.H.2    Schmidt, A.3
  • 37
    • 14544290953 scopus 로고    scopus 로고
    • A posteriori error analysis for time-dependent Ginzburg-Landau type equations
    • S. Bartels, A posteriori error analysis for time-dependent Ginzburg-Landau type equations, Numer Math 99 (2005), 557-583.
    • (2005) Numer Math , vol.99 , pp. 557-583
    • Bartels, S.1
  • 38
    • 23944495649 scopus 로고    scopus 로고
    • A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow
    • X. Feng, and, H.-J. Wu, A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow, J Sci Comput 24 (2005), 121-146.
    • (2005) J Sci Comput , vol.24 , pp. 121-146
    • Feng, X.1    Wu, H.-J.2
  • 39
    • 0032669875 scopus 로고    scopus 로고
    • On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors
    • S. Prudhomme, and, J. T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput Methods Appl Mech Engrg 176 (1999), 313-331.
    • (1999) Comput Methods Appl Mech Engrg , vol.176 , pp. 313-331
    • Prudhomme, S.1    Oden, J.T.2
  • 40
    • 0034817636 scopus 로고    scopus 로고
    • Goal-oriented error estimation and adaptivity for the finite element method
    • J. T. Oden, and, S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method, Comput Math Appl 41 (2001), 735-756.
    • (2001) Comput Math Appl , vol.41 , pp. 735-756
    • Oden, J.T.1    Prudhomme, S.2
  • 41
    • 0036865855 scopus 로고    scopus 로고
    • Estimation of modeling error in computational mechanics
    • J. T. Oden, and, S. Prudhomme, Estimation of modeling error in computational mechanics, J Comput Phys 182 (2002), 496-515.
    • (2002) J Comput Phys , vol.182 , pp. 496-515
    • Oden, J.T.1    Prudhomme, S.2
  • 42
    • 0442291345 scopus 로고    scopus 로고
    • Computable error estimators and adaptive techniques for fluid flow problems
    • T. J. Barth and H. Deconinck, editors Lecture Notes in Computational Science and Engineering, Springer-Verlag, Heidelberg
    • S. Prudhomme, and, J. T. Oden, Computable error estimators and adaptive techniques for fluid flow problems, Error estimation and adaptive discretization methods in computational fluid dynamics, T. J. Barth, and, H. Deconinck, editors, Vol. 25: Lecture Notes in Computational Science and Engineering, Springer-Verlag, Heidelberg, 2003, pp. 207-268.
    • (2003) Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics , vol.25 , pp. 207-268
    • Prudhomme, S.1    Oden, J.T.2
  • 43
    • 0030383153 scopus 로고    scopus 로고
    • A feed-back approach to error control in finite element methods: Basic analysis and examples
    • R. Becker, and, R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J Numer Math 4 (1996), 237-264.
    • (1996) East-West J Numer Math , vol.4 , pp. 237-264
    • Becker, R.1    Rannacher, R.2
  • 44
    • 85022001969 scopus 로고    scopus 로고
    • An optimal control approach to a posteriori error estimation in finite element methods
    • R. Becker, and, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer 10 (2001), 1-102. (Pubitemid 33305809)
    • (2001) ACTA NUMERICA , pp. 1-102
    • Becker, R.1    Rannacher, R.2
  • 45
    • 10444275414 scopus 로고    scopus 로고
    • Adaptive Finite Element Methods for Differential Equations
    • ETH Zürich, Birkhäuser.
    • W. Bangerth, and, R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2003.
    • (2003) Lectures in Mathematics
    • Bangerth, W.1    Rannacher, R.2
  • 46
    • 0036347672 scopus 로고    scopus 로고
    • Hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems
    • P. Houston, and, E. Süli, hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems, SIAM J Sci Comput 23 (2001), 1226-1252.
    • (2001) SIAM J Sci Comput , vol.23 , pp. 1226-1252
    • Houston, P.1    Süli, E.2
  • 47
    • 0012571382 scopus 로고    scopus 로고
    • Adaptive finite element approximation of hyperbolic problems
    • T. J. Barth and H. Deconinck, editors Lecture Notes in Computational Science and Engineering, Springer-Verlag, Heidelberg
    • E. Süli, and, P. Houston, Adaptive finite element approximation of hyperbolic problems, Error estimation and adaptive discretization methods in computational fluid dynamics, T. J. Barth, and, H. Deconinck, editors, Vol. 25: Lecture Notes in Computational Science and Engineering, Springer-Verlag, Heidelberg, 2003, pp. 269-344.
    • (2003) Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics , vol.25 , pp. 269-344
    • Süli, E.1    Houston, P.2
  • 48
    • 0031443748 scopus 로고    scopus 로고
    • A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations
    • M. Paraschivoiu, J. Peraire, and, A. T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comput Methods Appl Mech Engrg 150 (1997), 289-312.
    • (1997) Comput Methods Appl Mech Engrg , vol.150 , pp. 289-312
    • Paraschivoiu, M.1    Peraire, J.2    Patera, A.T.3
  • 49
    • 27744442426 scopus 로고    scopus 로고
    • A general Lagrangian formulation for the computation of a posteriori finite element bounds
    • T. J. Barth and H. Deconinck, editors Lecture Notes in Computational Science and Engineering, Springer-Verlag, Heidelberg
    • A. T. Patera, and, J. Peraire, A general Lagrangian formulation for the computation of a posteriori finite element bounds, Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, T. J. Barth, and, H. Deconinck, editors, Vol. 25: Lecture Notes in Computational Science and Engineering, Springer-Verlag, Heidelberg, 2003, pp. 159-206.
    • (2003) Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics , vol.25 , pp. 159-206
    • Patera, A.T.1    Peraire, J.2
  • 50
    • 0033309866 scopus 로고    scopus 로고
    • Finite element approximation of the acoustic wave equation: Error control and mesh adaptation
    • W. Bangerth, and, R. Rannacher, Finite element approximation of the acoustic wave equation: Error control and mesh adaptation, East-West J Numer Math 7 (1999), 263-282.
    • (1999) East-West J Numer Math , vol.7 , pp. 263-282
    • Bangerth, W.1    Rannacher, R.2
  • 51
    • 33744913843 scopus 로고    scopus 로고
    • Extensions of goal-oriented error estimation methods to simulations of highly-nonlinear response of shock-loaded elastomer-reinforced structures
    • D. Fuentes, D. Littlefield, J. T. Oden, and, S. Prudhomme, Extensions of goal-oriented error estimation methods to simulations of highly-nonlinear response of shock-loaded elastomer-reinforced structures, Comput Methods Appl Mech Engrg 195 (2006), 4659-4680.
    • (2006) Comput Methods Appl Mech Engrg , vol.195 , pp. 4659-4680
    • Fuentes, D.1    Littlefield, D.2    Oden, J.T.3    Prudhomme, S.4
  • 52
    • 33846680918 scopus 로고    scopus 로고
    • Goal-oriented error estimation for transient parabolic problems
    • P. Díez, and, G. Calderõn, Goal-oriented error estimation for transient parabolic problems, Comput Mech 39 (2007), 631-646.
    • (2007) Comput Mech , vol.39 , pp. 631-646
    • Díez, P.1    Calderõn, G.2
  • 53
    • 55549141189 scopus 로고    scopus 로고
    • Adaptivity with dynamic meshes for space-time finite element discretization of parabolic equations
    • M. Schmich, and, B. Vexler, Adaptivity with dynamic meshes for space-time finite element discretization of parabolic equations, SIAM J Sci Comput 30 (2008), 369-393.
    • (2008) SIAM J Sci Comput , vol.30 , pp. 369-393
    • Schmich, M.1    Vexler, B.2
  • 54
    • 68949186010 scopus 로고    scopus 로고
    • A space-time adaptive finite element algorithm based on dual weighted residual methodology for parabolic equations
    • R. Bermejo, and, J. Carpio, A space-time adaptive finite element algorithm based on dual weighted residual methodology for parabolic equations, SIAM J Sci Comput 31 (2009), 3324-3355.
    • (2009) SIAM J Sci Comput , vol.31 , pp. 3324-3355
    • Bermejo, R.1    Carpio, J.2
  • 56
    • 0026106415 scopus 로고
    • Adaptive finite element methods for parabolic problems I: A linear model problem
    • K. Eriksson, and, C. Johnson, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J Numer Anal 28 (1991), 43-77.
    • (1991) SIAM J Numer Anal , vol.28 , pp. 43-77
    • Eriksson, K.1    Johnson, C.2
  • 58
    • 84904150653 scopus 로고
    • Adaptive finite element methods for parabolic problems IV: Nonlinear problems
    • K. Eriksson, and, C. Johnson, Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J Numer Anal 32 (1995), 1729-1749.
    • (1995) SIAM J Numer Anal , vol.32 , pp. 1729-1749
    • Eriksson, K.1    Johnson, C.2
  • 59
    • 67649809189 scopus 로고
    • Adaptive finite element methods for parabolic problems V: Long-time integration
    • K. Eriksson, and, C. Johnson, Adaptive finite element methods for parabolic problems V: Long-time integration, SIAM J Numer Anal 32 (1995), 1750-1763.
    • (1995) SIAM J Numer Anal , vol.32 , pp. 1750-1763
    • Eriksson, K.1    Johnson, C.2
  • 61
    • 33749247298 scopus 로고    scopus 로고
    • Adaptive computational methods for parabolic problems
    • E. Stein, R. de Borst, and T. J. R. Hughes, editors Fundamentals, chapter 24, John Wiley & Sons, N.Y.
    • K. Eriksson, C. Johnson, and, A. Logg, Adaptive computational methods for parabolic problems, Encyclopedia of computational mechanics, E. Stein, R. de Borst, and, T. J. R. Hughes, editors, Vol. 1: Fundamentals, chapter 24, John Wiley & Sons, N.Y., 2004, pp. 675-702.
    • (2004) Encyclopedia of Computational Mechanics , vol.1 , pp. 675-702
    • Eriksson, K.1    Johnson, C.2    Logg, A.3
  • 62
  • 63
    • 0013027276 scopus 로고    scopus 로고
    • Models for phase separation and their mathematics
    • P. Fife, Models for phase separation and their mathematics, Electron J Differential Equations 48 (2000), 1-26.
    • (2000) Electron J Differential Equations , vol.48 , pp. 1-26
    • Fife, P.1
  • 64
    • 0003448545 scopus 로고    scopus 로고
    • Vol.: Graduate Studies in Mathematics, American Mathematical Society, Providence.
    • L. Evans, Partial Differential Equations, Vol. 19: Graduate Studies in Mathematics, American Mathematical Society, Providence, 1998.
    • (1998) Partial Differential Equations , vol.19
    • Evans, L.1
  • 66
    • 0002526735 scopus 로고
    • A mixed finite element method for the biharmonic equation
    • editor, Academic Press, New York, pp. Proceedings of the Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations held in Madison, Wisconsin, April 1-3, 1974.
    • P. G. Ciarlet, and, P.-A. Raviart, A mixed finite element method for the biharmonic equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, editor, Academic Press, New York, 1974, pp. 125-145. Proceedings of the Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations held in Madison, Wisconsin, April 1-3, 1974.
    • (1974) Mathematical Aspects of Finite Elements in Partial Differential Equations , pp. 125-145
    • Ciarlet, P.G.1    Raviart, P.-A.2    De Boor, C.3
  • 68
    • 85021895182 scopus 로고    scopus 로고
    • Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality
    • M. B. Giles, and, E. Süli, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numer (2002), 145-236.
    • (2002) Acta Numer , pp. 145-236
    • Giles, M.B.1    Süli, E.2
  • 70
    • 77950809885 scopus 로고    scopus 로고
    • Goal-oriented error estimation and adaptivity for free-boundary problems: The domain-map linearization approach
    • K. G. van der Zee, E. H. van Brummelen, and, R. de Borst, Goal-oriented error estimation and adaptivity for free-boundary problems: The domain-map linearization approach, SIAM J Sci Comput 32 (2010), 1064-1092.
    • (2010) SIAM J Sci Comput , vol.32 , pp. 1064-1092
    • Van Der Zee, K.G.1    Van Brummelen, E.H.2    De Borst, R.3
  • 71
    • 77950838818 scopus 로고    scopus 로고
    • Goal-oriented error estimation and adaptivity for free-boundary problems: The shape linearization approach
    • K. G. van der Zee, E. H. van Brummelen, and, R. de Borst, Goal-oriented error estimation and adaptivity for free-boundary problems: The shape linearization approach, SIAM J Sci Comput 32 (2010), 1093-1118.
    • (2010) SIAM J Sci Comput , vol.32 , pp. 1093-1118
    • Van Der Zee, K.G.1    Van Brummelen, E.H.2    De Borst, R.3
  • 72
    • 0002885640 scopus 로고    scopus 로고
    • A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems
    • D. Kröner, M. Ohlberger, and C. Rohde, editors Lecture Notes in Computational Science and Engineering, Springer, Berlin
    • E. Süli, A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems, An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger, and, C. Rohde, editors, Vol. 5: Lecture Notes in Computational Science and Engineering, Springer, Berlin, 1999, pp. 123-194.
    • (1999) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws , vol.5 , pp. 123-194
    • Süli, E.1
  • 73
    • 0034424235 scopus 로고    scopus 로고
    • A posteriori error estimation and adaptivity for degenerate parabolic problems
    • R. H. Nochetto, A. Schmidt, and, C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math Comp 69 (1999), 1-24.
    • (1999) Math Comp , vol.69 , pp. 1-24
    • Nochetto, R.H.1    Schmidt, A.2    Verdi, C.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.