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Can FEniCS solve a 1D PDE as the boundary of a 2D PDE?

+3 votes

We wish to solve a 2D ADR equation on a unit square, where the lower boundary (i.e. at y = 0) is itself a simple 1D PDE. For example,

$$ \frac{\partial u}{\partial t} = \nabla \cdot (\vec{a} u - D \nabla u) - ku $$

is defined on the whole domain, where $D$ and $k$ are constants, and the boundary conditions are mixed. Additionally, we have a separate PDE that must be solved, which is only defined on the lower boundary:

$$ \frac{\partial v}{\partial t} = u(\beta - v) - \alpha v $$

where $\alpha$ and $\beta$ are constants.

What we are wondering is: can FEniCS solve this 1D problem with the same mesh from the above 2D problem?

Any help is appreciated!

asked Oct 28, 2014 by mstobb FEniCS Novice (180 points)

1 Answer

+1 vote
 
Best answer

Yes, provided that you extract the 1D Boundary/SubMesh from the 2D mesh and that you solve the two problems iteratively. Take a look at the manifolds functionality in FEniCS, as described for instance here:

http://www.geosci-model-dev.net/6/2099/2013/gmd-6-2099-2013.html

answered Oct 30, 2014 by Marie E. Rognes FEniCS User (5,380 points)
selected Nov 3, 2014 by mstobb

Thanks for the answer and the reference. I'll definitely take a look at it.

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