Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid
Authors: Barrett John , Garcke Harald , Nürnberg Robert 
Published: November 2005

We consider a fully practical finite element approximation of the degenerate Cahn–Hilliard equation with elasticity: Find the conserved order parameter,

θ
(
x
,
t
)
∈
[
−
1
,
1
]

\theta (x,t)\in [-1,1]

, and the displacement field,

u
_

(
x
,
t
)
∈

R

2

\underline {u}(x,t) \in \mathbb {R}^2

, such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column gamma StartFraction partial-differential theta Over partial-differential t EndFraction 2nd Column a m p semicolon equals nabla period left-parenthesis b left-parenthesis theta right-parenthesis nabla left-bracket minus gamma normal upper Delta theta plus gamma Superscript negative 1 Baseline normal upper Psi prime left-parenthesis theta right-parenthesis plus one half c prime left-parenthesis theta right-parenthesis script upper C ModifyingBelow Below ModifyingBelow script upper E With bar With bar left-parenthesis u underbar right-parenthesis colon ModifyingBelow Below ModifyingBelow script upper E With bar With bar left-parenthesis u underbar right-parenthesis right-bracket right-parenthesis comma nabla period left-parenthesis c left-parenthesis theta right-parenthesis script upper C ModifyingBelow Below ModifyingBelow script upper E With bar With bar left-parenthesis u underbar right-parenthesis right-parenthesis 3rd Column a m p semicolon equals ModifyingBelow 0 With bar comma EndLayout">

γ

∂
θ

∂
t

a
m
p
;
=
∇

.

(
b
(
θ
)

∇

[
−
γ

Δ
θ
+

γ

−
1

Ψ
′

(
θ
)
+

1
2

c
′

(
θ
)

C

E

_

_

(

u
_

)
:

E

_

_

(

u
_

)
]

)

,
∇

.

(
c
(
θ
)

C

E

_

_

(

u
_

)
)

a
m
p
;
=

0
_

,

\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

θ
0

(
⋅
)
∈
[
−
1
,
1
]

\theta ^0(\cdot ) \in [-1,1]

on

θ
\theta

and boundary conditions on both equations. Here

γ
∈

R

>
0

\gamma \in {\mathbb R}_{>0}

is the interfacial parameter,

Ψ
\Psi

is a non-smooth double well potential,

E

_

_

\underline {\underline {\mathcal {E}}}

is the symmetric strain tensor,

C

\mathcal {C}

is the possibly anisotropic elasticity tensor,

c
(
s
)
:=

c
0

+

1
2

(
1
−

c
0

)

(
1
+
s
)

c(s):=c_0+\textstyle \frac 12\,(1-c_0)\,(1+s)

with

c
0

(
γ
)
∈

R

>
0

c_0(\gamma )\in {\mathbb R}_{>0}

and

b
(
s
)
:=
1
−

s
2

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.