Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 977-1002.

This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter (see e.g., Grasselli and Pata [Adv. Math. Sci. Appl. 13 (2003) 443–459, Comm. Pure Appl. Anal. 3 (2004) 849–881], Grasselli et al. [Comm. Pure Appl. Anal. 5 (2006) 827–838], Wu et al. [Math. Models Methods Appl. Sci. 17 (2007) 125–153, J. Math. Anal. Appl. 329 (2007) 948–976]). On the other hand, a time discretization of an initial value problem for an abstract evolution equation has been studied (see e.g., Colli and Favini [Int. J. Math. Math. Sci. 19 (1996) 481–494]) and Schimperna [J. Differ. Equ. 164 (2000) 395–430] has established existence of solutions to an abstract problem applying to a nonlinear phase-field system of Caginalp type on a bounded domain by employing a time discretization scheme. In this paper we focus on a time discretization of a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. Moreover, we can establish an error estimate for the difference between continuous and discrete solutions.

DOI : 10.1051/m2an/2019086
Classification : 35A35, 47N20, 35G30, 35L70
Mots-clés : Simultaneous abstract evolution equations, parabolic-hyperbolic phase-field systems, existence, time discretizations, error estimates
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     title = {Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {977--1002},
     publisher = {EDP-Sciences},
     volume = {54},
     number = {3},
     year = {2020},
     doi = {10.1051/m2an/2019086},
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     zbl = {1437.65120},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2019086/}
}
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Kurima, Shunsuke. Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 977-1002. doi : 10.1051/m2an/2019086. http://www.numdam.org/articles/10.1051/m2an/2019086/

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