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Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid


Authors: John W. Barrett, Harald Garcke and Robert Nürnberg
Journal: Math. Comp. 75 (2006), 7-41
MSC (2000): Primary 65M60, 65M12, 65M50, 35K55, 35K65, 35K35, 82C26, 74F15
DOI: https://doi.org/10.1090/S0025-5718-05-01802-8
Published electronically: October 12, 2005
MathSciNet review: 2176388
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Abstract: We consider a fully practical finite element approximation of the degenerate Cahn-Hilliard equation with elasticity: Find the conserved order parameter, $ \theta(x,t)\in[-1,1]$, and the displacement field, $ \underline{u}(x,t) \in \mathbb{R}^2$, such that

$\displaystyle \gamma\,\textstyle{\frac{\partial \theta}{\partial t}}$ $\displaystyle = \nabla\,.\, (b(\theta) \,\nabla \,[- \gamma \, \Delta \theta + ... ...line{u}): \underline{\underline{\mathcal{E}}} (\underline{u})]\,) \,, \nonumber$    
$\displaystyle \nabla \,.\, (c(\theta) \,\mathcal{C}\, \underline{\underline{\mathcal{E}}} (\underline{u}) )$ $\displaystyle = \underline{0}\,,$    

subject to an initial condition $ \theta^0(\cdot) \in [-1,1]$ on $ \theta$ and boundary conditions on both equations. Here $ \gamma \in {\mathbb{R}}_{>0} $ is the interfacial parameter, $ \Psi$ is a non-smooth double well potential, $ \underline{\underline{\mathcal{E}}} $ is the symmetric strain tensor, $ \mathcal{C}$ is the possibly anisotropic elasticity tensor, $ c(s):=c_0+\textstyle\frac12\,(1-c_0)\,(1+s)$ with $ c_0(\gamma)\in {\mathbb{R}}_{>0}$ and $ b(s):=1-s^2$ is the degenerate diffusional mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.


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Additional Information

John W. Barrett
Affiliation: Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom
Email: j.barrett@imperial.ac.uk

Harald Garcke
Affiliation: NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: harald.garke@mathematik.uni-regensburg.de

Robert Nürnberg
Affiliation: Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom
Email: robert.nurnberg@imperial.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-05-01802-8
Received by editor(s): April 21, 2004
Received by editor(s) in revised form: January 26, 2005
Published electronically: October 12, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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