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-   -   Domain decomposition and Ax=b?? (http://www.cfd-online.com/Forums/openfoam-solving/86673-domain-decomposition-ax-b.html)

pkr March 29, 2011 17:17

Domain decomposition and Ax=b??
 
I am trying to understand the way Ax=b is solved in parallel across the processors.

The original mesh is comprised of 5 cells. The original matrix A without any decomposition is:
-4 0 4 0 0
0 -3 0 0 3
4 0 -8 9 0
0 0 9 -6 5
0 3 0 5 -7

After using decomposePar utility the matrix A is decomposed in two smaller matrices as follows:
First matrix
========
-3 0 3
0 -6 5
3 5 -7

Second Matrix
==========
-4 4
4 8


When PCG solver is invoked, each process solves Ax=b independently and exchanges some information with other process to ensure correctness.

My queries are as follows:

1. What is being exchanged? I see that in every iteration during Amul, some parts of vector pA is being sent to neighboring process. The neighboring process receives pA and performs additional computation to rectify wA.

wA[] -= coeffs[] * pA_received[];

2. Where do the coefficients come from? What's the significance of coeffs?

3. How does subtracting "coeffs[] * pA_received[]" from wA[] leads to the correct solution mathematically?


Thanks in Advance.

sleepdeprivation March 30, 2011 08:31

I'm not totally sure how OF does it, but from what you've posted, it looks like its doing a Jacobi iteration, but changing the order of the matrix in the decompose step. (or a red/black Gauss-Seidel, which is similar)

Matrix 1 is the rows of 2,4,&5 while matrix 2 is the 1 & 3 rows.

So it updates with the local matrix, then takes the values of the cells outside the domain to finish the update. Thus coeffs[] would be the coefficients of the original matrix that relate the pole to neighbors in another domain.


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