Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid
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- by John W. Barrett, Harald Garcke and Robert Nürnberg;
- Math. Comp. 75 (2006), 7-41
- DOI: https://doi.org/10.1090/S0025-5718-05-01802-8
- Published electronically: October 12, 2005
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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 \begin{align*} \gamma \textstyle {\frac {\partial \theta }{\partial t}} & = \nabla . (b(\theta ) \nabla [- \gamma \Delta \theta + \gamma ^{-1} \Psi ’(\theta ) + \textstyle \frac 12 c’(\theta ) \mathcal {C} \underline {\underline {\mathcal {E}}} (\underline {u}): \underline {\underline {\mathcal {E}}} (\underline {u})] ) , \nonumber \nabla . (c(\theta ) \mathcal {C} \underline {\underline {\mathcal {E}}} (\underline {u}) ) & = \underline {0} , \end{align*} 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, $c(s):=c_0+\textstyle \frac 12 (1-c_0) (1+s)$ 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.References
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Bibliographic Information
- John W. Barrett
- Affiliation: Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom
- MR Author ID: 31635
- Email: j.barrett@imperial.ac.uk
- Harald Garcke
- Affiliation: NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany
- MR Author ID: 352477
- Email: harald.garke@mathematik.uni-regensburg.de
- Robert Nürnberg
- Affiliation: Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom
- MR Author ID: 698349
- Email: robert.nurnberg@imperial.ac.uk
- Received by editor(s): April 21, 2004
- Received by editor(s) in revised form: January 26, 2005
- Published electronically: October 12, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2176388