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such that the partial differential equation is then defined as
∂c
∂t
=
M
c
∇
ʴ
F
CH
ʴc
∇·
M
c
∇
μ
c
−
ˇ
ʔc
,
1
=
∇·
(4)
where
M
c
=
c
(1
c
) represents a mobility. Finally, the strong form of this
equation, given that concentration must be conserved [7], can be expressed as:
over the spatial domain
ʩ
and the time interval ]0
,T
[, given
c
0
:
ʩ
−
−ₒ
R
, find
c
:
ʩ
×
[0
,T
]
−ₒ
R
M
c
∇
μ
c
−
ˇ
ʔc
=0
⊧
⊨
∂c
∂t
−∇·
1
on
ʩ
×
]0
,T
]
(5)
⊩
c
=
c
0
on
ʩ
×{
t
=0
}
where
c
0
denotes the initial condition, the natural boundary conditions are taken
to be equal to zero, and periodic boundary conditions are considered in all di-
rections. To derive the weak form for this equation, we let
denote the trial and
weighting function spaces, and multiply the strong form (5) by a test function
w
V
and integrate by parts,. The problem using the Galerkin method is then
stated as: find
c
∈V
∈V
such that
∀
w
∈V
,
w,
∂c
∂t
+
M
c
ʔc
+
ʔw,
1
ˇ
M
c
ʔc
μ
c
+
1
∇
w,M
c
∇
ˇ
∇
=0
,
(6)
ʩ
ʩ
ʩ
2
inner product over the domain
ʩ
and
where (
., .
)
ʩ
represents the
L
V
needs to
2
represents the Sobolev space of square integrable
functions with square integrable first and second derivatives. We now discretize
the infinite dimensional problem in space, and derive the semidiscrete formula-
tion which can be stated as: find
c
h
2
-conforming, where
be
H
H
h
w
h
h
∈V
ↂV
such that
∀
∈V
ↂV
w
h
, c
h
ʩ
+
M
c
ʔc
h
+
ʔw
h
,
1
ˇ
M
c
ʔc
h
μ
c
+
1
w
h
,M
c
∇
∇
ˇ
∇
=0
.
(7)
ʩ
ʩ
h
is spanned by the linear combination of
We suppose that the discrete space
V
1
-continuous B-spline basis functions.
With regards to PetIGA, if one is able to get equation (7) and an initial
condition, testing of the residual can already be done to check if the system
converges to a solution, or compare the results to a benchmark problem. Being a
nonlinear time-dependent problem, the Cahn-Hilliard model requires the use of a
Jacobian if a Newton-type scheme [8] is used. Nonetheless, by being built on top
of PETSc, the user can use available functions to approximate the Jacobian. This
can save valuable time while prototyping and debugging code, as the residual
can be tested without explicitly coding the Jacobian [4]. With regards to the
time-discretization, we employ the adaptive scheme from [8], which uses the
generalized-
ʱ
method. By setting an initial condition
c
0
such that
c
(
t
=0
,
x
)=0
.
63 +
ʷ,
basis functions
N
A
,whichare
C
(8)
