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How can we define the initial value for Cahn-Hilliard problem (links) using FEM in MMA?

Complete test code (I am using MIXED formulation for C1 Problem):

Needs["NDSolve`FEM`"]
\[CapitalOmega]=ImplicitRegion[{0<=x<=1,0<=y<=1},{x,y}];
RegionPlot[\[CapitalOmega],PlotRange->{{0,1},{0,1}}]
Mobi = 1.0; lame = 0.01;
op1 = \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(u[t, x, y]\)\) - Mobi.\!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(v[t, x, y]\)\)

op2 = v[t, x, y] - 
  200 u[t, x, y] (1 - 3 u[t, x, y] + 2 u[t, x, y]^2) + lame.\!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[t, x, 
     y]\)\(\ \)\)

Subscript[\[CapitalGamma], N1] = 
 NeumannValue[Mobi* lame.u[t, x, y], {0 <= x <= 1, 0 <= y <= 1}]

{nufun, nvfun} = 
  NDSolveValue[{op1 == Subscript[\[CapitalGamma], N1], op2 == 0}, {u, 
    v}, {t, 0, 100}, {x, y} \[Element] \[CapitalOmega]];
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  • $\begingroup$ There are also other problems in your code: The Dots after Mobi and lame have to be removed and you have not specified boundary conditions for v. $\endgroup$
    – Henrik Schumacher
    Jul 20, 2019 at 1:27
  • $\begingroup$ @HenrikSchumacher I have checked the bcs, you may see Eq 5.1 from fenicsproject.org/olddocs/dolfin/1.3.0/python/demo/documented/… $\endgroup$
    – ABCDEMMM
    Jul 20, 2019 at 1:33
  • $\begingroup$ Yeah, so you have to specify them also in the call to NDSolve. $\endgroup$
    – Henrik Schumacher
    Jul 20, 2019 at 1:36

1 Answer 1

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This is one of many ways to create "random" initial value conditions:

Ωdisc = ToElementMesh[Ω];
n = Length[Ωdisc["Coordinates"]];
u0 = ElementMeshInterpolation[{Ωdisc}, RandomReal[{-2, 2}, n]];
Plot3D[u0[x, y], {x, y} ∈ Ωdisc]

It might be a good idea to use Ωdisc] also in the call to NDSolveValues instead of Ω.

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  • $\begingroup$ @Herik Schumacher, this initial value definition not works. $0.625107,0.634007,<42>>, 0.639777,0.627036,0.625157, \ll 3151>>\},\{\text { Automatic }\} ]$ should be a rank 2 tensor of machine-size real numbers. $\endgroup$
    – ABCDEMMM
    Jul 20, 2019 at 9:55
  • $\begingroup$ That's probably because this is only an initial condition for u, not for v. $\endgroup$
    – Henrik Schumacher
    Jul 20, 2019 at 12:37
  • $\begingroup$ u[0, x, y] == u0[x, y]??? $\endgroup$
    – ABCDEMMM
    Jul 20, 2019 at 12:40
  • $\begingroup$ Yes, that's how you should use it. But v needs also initial conditions. And boundary conditions. $\endgroup$
    – Henrik Schumacher
    Jul 20, 2019 at 12:43

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