Solving separate FEM problems on subdomains

[Corbin Foucart]

Hello all,

I have an extruded mesh, and I would like to solve a finite element problem separately on each 'column' as part of a larger finite element solve in which all DoF are coupled through a discontinuous Galerkin scheme for an elliptic equation--so multiply defined values on the boundaries of each column are not a problem.

• I already have the columns tagged by material ID
• I imagine I should instantiate a list of new Triangulations, DoFHandlers, and solver classes for each column and then map the solutions back to the original domain
• Is there a good way to extract a separate triangulation for each column? I was looking for something similar to extract_boundary_mesh but with a material_id
  

Alternatively, it seems that I could assign the DoFs in the globally coupled DoFHandler with DoFTools::get_subdomain_association(). Would this be a better way to set up each finite element problem by column? It's unclear to me if I could use that information to set up an independent linear system for each subdomain.

Any guidance would be appreciated!

Corbin

[Bruno Turcksin]

Corbin,

Creating multiple Triangulations and mapping them is not easy to do. Instead, you want to use FE_Nothing to ignore some cells. Take a loot at step-46 https://dealii.org/current/doxygen/deal.II/step_46.html

Best,

Bruno

[Wolfgang Bangerth]

On 3/21/22 16:47, Corbin Foucart wrote:
>
> I have an extruded mesh, and I would like to solve a finite element problem
> separately on each 'column' as part of a larger finite element solve in which
> all DoF are coupled through a discontinuous Galerkin scheme for an elliptic
> equation--so multiply defined values on the boundaries of each column are not
> a problem.
>

> * I already have the columns tagged by material ID
> * I imagine I should instantiate a list of new Triangulations, DoFHandlers,

> and solver classes for each column and then map the solutions back to the
> original domain

> * *Is there a good way to extract a separate triangulation for each column*?

> I was looking for something similar to extract_boundary_mesh but with a
> material_id
>
> Alternatively, it seems that I could assign the DoFs in the globally coupled

> DoFHandler with DoFTools::get_subdomain_association(). *Would this be a better
> way to set up each finite element problem by column?* It's unclear to me if I

> could use that information to set up an independent linear system for each
> subdomain.

Corbyn,
you could go the way of creating triangulations, DoFHandlers, etc, for each of
the columns. But you might also want to consider one of the following two
approaches:

* Since you're already doing a DG approach, consider building your matrix with
only the interface terms that correspond to the "bottom" and "top" of each of
your cells. This means that you don't build any term that corresponds to
coupling columns to each other. Solving such a linear system then corresponds
to solving all of the 1d problems at the same time. (I will note that such a
matrix would also make for a good preconditioner; it corresponds to one of the
directions one would solve for in the ADI method.)

Conceptually, the matrix you obtain using the approach above is block diagonal
where each block corresponds to one column. In practice, unless you do
something special, the matrix will not actually look block diagonal because
the order of DoFs does not respect the column-by-column order. But it wouldn't
be very difficult to implement a function that renumbers DoFs in such a way
that they are numbered column by column, and then you would indeed have a
block diagonal matrix.

* If you don't like this approach, you can create a mapping from the DoFs of
each column to a column-local numbering from zero to n_dofs_per_column.
Instead of writing into global vectors and matrices in the method above where
you omit the lateral face terms, you can translate writing into smaller
matrices and vectors that correspond to just one column. You can then solve
the resulting linear system. The mapping of DoF indices then also allows you
to get back at what the global index of these DoFs was.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Corbin Foucart]

Bruno and Wolfgang,

Thank you for the helpful suggestions--I had not thought of the block-diagonal linear system approach in particular, but I think it is a clever way to do what I had previously planned. I don't anticipate it being too expensive, even unordered, but if so, I will implement one of the other two approaches.

Corbin

[Corbin Foucart]

I never replied to this, but the approach of using assembling only the interface terms worked very well, and was painless to code. 

Thanks for the great idea, this saved me a ton of time :-)

Corbin