Amit,
I am trying to solve a reaction-diffusion PDE with two reactants A and B.
Reactant B diffuses much faster than A i.e. it instantly diffuses to become
uniform over the whole domain in response to changes in concentration of A.
Thus, the concentration of this reactant is simply a real number that varies
with time but not with space. The total mass of reactants A and B is fixed.
The governing equations are as shown in the following image
System.png
Here, a(x, t) and b(t) are the concentrations of the reactants. D is the
diffusion co-efficient and f(a, b) is a non-linear reaction term. M is total
mass of A and B and S is the surface area of the domain. I came up with the
following weak form from these two equations
WeakForm.png
You can do it that way, but (i) that's awkward because now you have this
global term in your problem, and (ii) it's not really useful to think of a
scalar equation as one you wanted to solve with a "field" for 'b' and 'r'.
But let's pretend that you wanted to do it that way (and that for simplicity,
f(a,b) is a linear function of the two variables, so that your problem is
linear). Then the way I would approach this is by conceptually saying that you
have one DoFHandler for 'a', which is your field, and your overall linear
problem leads to a matrix that you can partition as follows:
[ A C ] [ a ] = [ ... ]
[ B^T I ] [ b ] = [ M/S ]
Here, C results from f(a,b) = X a + C b, and B^T is a 1xn vector of all ones.
I is the 1x1 unit matrix. A is what you get from the spatial discretization of
'a' plus the X that comes out of 'f'.
You can build this linear system as a BlockSparseMatrix, where only A is
associated with the DoFHandler.
I would imagine that a better way to solve this problem is to do some sort of
operator splitting where you alternate between updating 'a' and 'b'. The
equations for the updates are obvious.
Best
W.
--
------------------------------------------------------------------------
Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google Groups "deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to dealii+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/dealii/cff5592c-feea-9514-7797-6be1356ce88a%40colostate.edu.
