data structures for efficient/easy implementation of finite volume method for 2D Poisson equation

Question

My question is about implementation alone.

• Consider a square domain with regular square, cell centred finite volumes. This is for the multiscale finite volume method (Jenny and Lunati)

• I need to solve the Poisson equation on each cell of the "dual mesh", i.e., the mesh constructed by joining centroids of each cell. (the dual mesh will look like a translation of the original mesh; 4 cells of the dual mesh make up one cell of primary mesh)

• I will need to use this to construct the "transmissibilities" for each cell of the primary mesh.

My question is:

1. What kind of date structure should I use that will help me go from primary to dual mesh easily
2. I am confused what I shld store, the coordinates of the cell centres, or the locations of the faces ?
3. How do I store the data that will tell me which 4 dual cells make up a given primary cell ?

It seems to me I will need to do a kind of "assembly" for my transmissibility matrix as I visit each dual cell and solve a local Poisson problem.

Answer 1

I'm not familiar with the work by Jenny and Lunati, so this answer might be wrong. But if I understand correctly, the geometry you want can be generated in a few lines using PyClaw; see this notebook. This figure shows the primary and dual grids:

You can also use mapped grids, and ghost cells are automatically generated if you need them:

If either of those is what you're looking for, I can post all the necessary code here inline (it's in the notebook linked above).

Answer 2

Did you consider implementing the algebraic MSFV instead? The implementation complexity is a lot less. Details can be found in the paper "An operator formulation of the multiscale finite-volume method with correction function" by Lunati, 2009.