April  2016, 9(2): 537-556. doi: 10.3934/dcdss.2016011

Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex

2. 

University "Al. I. Cuza" of Iasi, 700506 Iaşi, Romania

Received  August 2014 Revised  November 2014 Published  March 2016

The paper concerns with the existence, uniqueness, regularity and the approximation of solutions to the nonlinear phase-field (Allen-Cahn) equation, endowed with non-homogeneous dynamic boundary conditions (depending both on time and space variables). It extends the already studied types of boundary conditions, which makes the problem to be more able to describe many important phenomena of two-phase systems, in particular, the interactions with the walls in confined systems. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual numerical algorithm is formulated in the end.
Citation: Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011
References:
[1]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta Metall., 27 (1979), 1084.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system,, Intern. J. Com. Math., 62 (1996), 209.  doi: 10.1080/00207169608804538.  Google Scholar

[3]

T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions,, Numer. Funct. Anal. and Optimiz., 30 (2009), 199.  doi: 10.1080/01630560902841120.  Google Scholar

[4]

T. Benincasa, A. Favini and C. Moroşanu, A Product Formula Approach to a Non-homogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System. PART I: A Phase-field Model,, J. Optim. Theory and Appl., 148 (2011), 14.  doi: 10.1007/s10957-010-9742-x.  Google Scholar

[5]

J. L. Boldrini, B. M. C. Caretta and E. Fernández-Cara, Analysis of a two-phase field model for the solidification of an alloy,, J. Math. Anal. Appl., 357 (2009), 25.  doi: 10.1016/j.jmaa.2009.03.063.  Google Scholar

[6]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits,, Euro. Jnl of Applied Mathematics, 9 (1998), 417.  doi: 10.1017/S0956792598003520.  Google Scholar

[7]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions,, Nonlinear Analysis: Theory, 79 (2013), 12.  doi: 10.1016/j.na.2012.11.010.  Google Scholar

[8]

C. Cavaterra, C. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions,, Nonlinear Anal. TMA, 72 (2010), 2375.  doi: 10.1016/j.na.2009.11.002.  Google Scholar

[9]

L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase field system with dynamic boundary conditions and singular potentials,, J. Math. Anal. Appl., 343 (2008), 557.  doi: 10.1016/j.jmaa.2008.01.077.  Google Scholar

[10]

L. Cherfils, S. Gatti and A. Miranville, Long time behavior to the Caginalp system with singular potentials and dynamic boundary conditions,, Commun. Pure Appl. Anal., 11 (2012), 2261.  doi: 10.3934/cpaa.2012.11.2261.  Google Scholar

[11]

M. Conti, S. Gatti and A. Miranville, Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions,, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 485.  doi: 10.3934/dcdss.2012.5.485.  Google Scholar

[12]

M. Conti, S. Gatti and A. Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions,, Anal. Appl. (Singap.), 11 (2013).   Google Scholar

[13]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in Internat. Ser. Numer. Math., 95 (1990), 46.   Google Scholar

[14]

I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Clarendon, (1995).   Google Scholar

[15]

C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions,, Discrete Contin. Dyn. Syst., 22 (2008), 1009.  doi: 10.3934/dcds.2008.22.1009.  Google Scholar

[16]

C. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 689.  doi: 10.3934/cpaa.2009.8.689.  Google Scholar

[17]

C. Gal, M. Grasselli, A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535.  doi: 10.1007/s00030-008-7029-9.  Google Scholar

[18]

C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions,, Nonlinear phenomena with energy dissipation, 29 (2008), 117.   Google Scholar

[19]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, Differential equations: inverse and direct problems, 251 (2006), 149.  doi: 10.1201/9781420011135.ch9.  Google Scholar

[20]

K.-H. Hoffman and L. Jiang, Optimal control problem of a phase field model for solidification,, Numer. Funct. Anal. and Optimiz., 13 (1992), 11.  doi: 10.1080/01630569208816458.  Google Scholar

[21]

Gh. Iorga, C. Moroşanu and I. Tofan, Numerical simulation of the thickness accretions in the secondary cooling zone of a continuous casting machine,, Metalurgia International, XIV (2009), 72.   Google Scholar

[22]

H. Israel, Long time behavior of an Allen-Cahn type equation with singular potential and dynamic boundary conditions,, Journal of Applied Analysis and Computation, 2 (2012), 29.   Google Scholar

[23]

N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems,, Nonlinear Anal. TMA, 22 (1994), 1163.  doi: 10.1016/0362-546X(94)90235-6.  Google Scholar

[24]

O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type,, Prov. Amer. Math. Soc., (1968).   Google Scholar

[25]

J. L. Lions, Control of Distributed Singular Systems,, Gauthier-Villars, (1985).   Google Scholar

[26]

A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions,, Appl. Math. Model., 40 (2016), 192.  doi: 10.1016/j.apm.2015.04.039.  Google Scholar

[27]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709.  doi: 10.1002/mma.590.  Google Scholar

[28]

C. Moroşanu, Approximation of the phase-field transition system via fractional steps method,, Numer. Funct. Anal. and Optimiz., 18 (1997), 623.  doi: 10.1080/01630569708816782.  Google Scholar

[29]

C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods,, Bentham Science Publishers, (2012).  doi: 10.2174/97816080535061120101.  Google Scholar

[30]

C. Moroşanu and D. Motreanu, A generalized phase field system,, J. Math. Anal. Appl., 237 (1999), 515.  doi: 10.1006/jmaa.1999.6467.  Google Scholar

[31]

C. Moroşanu and D. Motreanu, Uniqueness and approximation for the phase field equation in caginalp's model,, Intern. J. of Appl. Math., 2 (2000), 113.   Google Scholar

[32]

C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity,, International Journal of Differential Equations and Applications, 1 (2000), 187.   Google Scholar

[33]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions,, Phys. D., 43 (1990), 44.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

show all references

References:
[1]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta Metall., 27 (1979), 1084.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system,, Intern. J. Com. Math., 62 (1996), 209.  doi: 10.1080/00207169608804538.  Google Scholar

[3]

T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions,, Numer. Funct. Anal. and Optimiz., 30 (2009), 199.  doi: 10.1080/01630560902841120.  Google Scholar

[4]

T. Benincasa, A. Favini and C. Moroşanu, A Product Formula Approach to a Non-homogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System. PART I: A Phase-field Model,, J. Optim. Theory and Appl., 148 (2011), 14.  doi: 10.1007/s10957-010-9742-x.  Google Scholar

[5]

J. L. Boldrini, B. M. C. Caretta and E. Fernández-Cara, Analysis of a two-phase field model for the solidification of an alloy,, J. Math. Anal. Appl., 357 (2009), 25.  doi: 10.1016/j.jmaa.2009.03.063.  Google Scholar

[6]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits,, Euro. Jnl of Applied Mathematics, 9 (1998), 417.  doi: 10.1017/S0956792598003520.  Google Scholar

[7]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions,, Nonlinear Analysis: Theory, 79 (2013), 12.  doi: 10.1016/j.na.2012.11.010.  Google Scholar

[8]

C. Cavaterra, C. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions,, Nonlinear Anal. TMA, 72 (2010), 2375.  doi: 10.1016/j.na.2009.11.002.  Google Scholar

[9]

L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase field system with dynamic boundary conditions and singular potentials,, J. Math. Anal. Appl., 343 (2008), 557.  doi: 10.1016/j.jmaa.2008.01.077.  Google Scholar

[10]

L. Cherfils, S. Gatti and A. Miranville, Long time behavior to the Caginalp system with singular potentials and dynamic boundary conditions,, Commun. Pure Appl. Anal., 11 (2012), 2261.  doi: 10.3934/cpaa.2012.11.2261.  Google Scholar

[11]

M. Conti, S. Gatti and A. Miranville, Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions,, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 485.  doi: 10.3934/dcdss.2012.5.485.  Google Scholar

[12]

M. Conti, S. Gatti and A. Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions,, Anal. Appl. (Singap.), 11 (2013).   Google Scholar

[13]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in Internat. Ser. Numer. Math., 95 (1990), 46.   Google Scholar

[14]

I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Clarendon, (1995).   Google Scholar

[15]

C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions,, Discrete Contin. Dyn. Syst., 22 (2008), 1009.  doi: 10.3934/dcds.2008.22.1009.  Google Scholar

[16]

C. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 689.  doi: 10.3934/cpaa.2009.8.689.  Google Scholar

[17]

C. Gal, M. Grasselli, A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535.  doi: 10.1007/s00030-008-7029-9.  Google Scholar

[18]

C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions,, Nonlinear phenomena with energy dissipation, 29 (2008), 117.   Google Scholar

[19]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, Differential equations: inverse and direct problems, 251 (2006), 149.  doi: 10.1201/9781420011135.ch9.  Google Scholar

[20]

K.-H. Hoffman and L. Jiang, Optimal control problem of a phase field model for solidification,, Numer. Funct. Anal. and Optimiz., 13 (1992), 11.  doi: 10.1080/01630569208816458.  Google Scholar

[21]

Gh. Iorga, C. Moroşanu and I. Tofan, Numerical simulation of the thickness accretions in the secondary cooling zone of a continuous casting machine,, Metalurgia International, XIV (2009), 72.   Google Scholar

[22]

H. Israel, Long time behavior of an Allen-Cahn type equation with singular potential and dynamic boundary conditions,, Journal of Applied Analysis and Computation, 2 (2012), 29.   Google Scholar

[23]

N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems,, Nonlinear Anal. TMA, 22 (1994), 1163.  doi: 10.1016/0362-546X(94)90235-6.  Google Scholar

[24]

O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type,, Prov. Amer. Math. Soc., (1968).   Google Scholar

[25]

J. L. Lions, Control of Distributed Singular Systems,, Gauthier-Villars, (1985).   Google Scholar

[26]

A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions,, Appl. Math. Model., 40 (2016), 192.  doi: 10.1016/j.apm.2015.04.039.  Google Scholar

[27]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709.  doi: 10.1002/mma.590.  Google Scholar

[28]

C. Moroşanu, Approximation of the phase-field transition system via fractional steps method,, Numer. Funct. Anal. and Optimiz., 18 (1997), 623.  doi: 10.1080/01630569708816782.  Google Scholar

[29]

C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods,, Bentham Science Publishers, (2012).  doi: 10.2174/97816080535061120101.  Google Scholar

[30]

C. Moroşanu and D. Motreanu, A generalized phase field system,, J. Math. Anal. Appl., 237 (1999), 515.  doi: 10.1006/jmaa.1999.6467.  Google Scholar

[31]

C. Moroşanu and D. Motreanu, Uniqueness and approximation for the phase field equation in caginalp's model,, Intern. J. of Appl. Math., 2 (2000), 113.   Google Scholar

[32]

C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity,, International Journal of Differential Equations and Applications, 1 (2000), 187.   Google Scholar

[33]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions,, Phys. D., 43 (1990), 44.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[1]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[2]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381

[3]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052

[4]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353

[5]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[6]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[7]

Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021009

[8]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[9]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[10]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380

[11]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[12]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[13]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[14]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061

[15]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083

[16]

Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283

[17]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[18]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[19]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

[20]

Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]