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Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. (English) Zbl 1437.65120
Summary: 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., [M. Grasselli and V. Pata, Adv. Math. Sci. Appl. 13, No. 2, 443–459 (2003; Zbl 1057.37068); Commun. Pure Appl. Anal. 3, No. 4, 849–881 (2004; Zbl 1079.35022); M. Grasselli et al., Commun. Pure Appl. Anal. 5, No. 4, 827–838 (2006; Zbl 1134.35017); H. Wu et al., Math. Models Methods Appl. Sci. 17, No. 1, 125–153 (2007; Zbl 1120.35024); J. Math. Anal. Appl. 329, No. 2, 948–976 (2007; Zbl 1154.35329)]). On the other hand, a time discretization of an initial value problem for an abstract evolution equation has been studied (see e.g., [P. Colli and A. Favini, Int. J. Math. Math. Sci. 19, No. 3, 481–494 (1996; Zbl 0859.35077)] and [G. Schimperna, J. Differ. Equations 164, No. 2, 395–430 (2000; Zbl 0978.35075)] 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.

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35A35 Theoretical approximation in context of PDEs 47N20 Applications of operator theory to differential and integral equations 35G30 Boundary value problems for nonlinear higher-order PDEs 35L70 Second-order nonlinear hyperbolic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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