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

Alain Miranville; Costică Moroşanu

doi:10.3934/dcdss.2016011

Article Contents

2016, Volume 9, Issue 2: 537-556. Doi: 10.3934/dcdss.2016011

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

Alain Miranville^1, and
Costică Moroşanu^2,

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

Received: July 31, 2014

Revised: October 31, 2014

Published: March 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 65N12, 80AXX.

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