[dune-fem] A dune-fem project for the thesis

[Tobias Malkmus]

Hi Mauro

Sorry to let your wait.

First of all, the problem can be solved within Dune-Fem context.
I assume ε is small, so your problem has different scaling properties.

In general Dune-Fem can solve equations of the form

∂tv = F( v )

with given initial and boundary conditions as you have mentioned.
The time discretisation in dune-Fem is a k-staged implict or explicit 
Runge-Kutta Method up to order 4.

Lets come to the question of space discretisation.
An example for an advection-diffusion equation
using a DG discretisation in space can be found in the 'actual' 
fem-howto. It solves an equation of the form

∂tv + \nabla (w*v) - ε div \mathbf{\nabla} v = 0

with given w, which is closed to your problem.

If you are interested into Lagrange elements, an example is not yet 
available.

Next week there will be a summerschool for Dune-Fem in Freiburg, i don't 
know wether registration time expired.
Solving advection-diffusion equation with dune-fem is one topic of this 
school.

If you want to attend this school you should ask Martin Nolte wether it 
is possible.

Best Tobias

On 09/28/2012 12:28 PM, Mauro Pagani wrote:
>
> Hi, I contact you because I want to implement a code of a model of a
> reaction-diffusion equations system of the form:
>
> ε∂tv + 4v(v*v-1) - ε*ε div Mi\mathbf{\nabla}ui = 0
>
> ε∂tv + 4v(v*v-1) - ε*ε div Me\mathbf{\nabla}ue = 0
>
> with
>
> Ω = (0,1.4) x (0,1.4)
>
> v = ui - ue
>
> ui(.,0)=-0.5 in A0= a quarter of a circumference of unit radius
> ui(.,0)=+0.5 in < font style="font-size:10pt" size="2">Ω\A0
> ue(.,0)=+0.5 in A0
> ue(.,0)=-0.5 in Ω\A0
>
> v =1 on Γ1 =top and right sides of the square
>
> ∂v/∂n =0 on Γ2 =left and bottom sides of the square
>
>
>
> This problem is referred to the cardiac bioelectric activity. I choose
> this topic for my thesis in Maths in Brescia (Italy).
>
> Could you help me with some documentation or ideas?..I see tutorials in
> dune-fem-howto, I understand the basis concepts, but a problem like this
> is a little hard to solve.
>
> Thank you very much
>
> Mauro Pagani
>
>
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> dune-fem mailing list
> dune-fem at dune-project.org
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>


-- 
Tobias Malkmus                 <tomalk at mathematik.uni-freiburg.de>

Mathematisches Institut               Tel: +49 (0) 761 203 5651
Abt. für Angewandte Mathematik        Universität Freiburg
Hermann-Herder-Str. 10
79104 Freiburg