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Poisson Eqn in a concentric cube



Hi,
 
     I would like to solve the Poisson equation (3D) in a concentric type of cube as in the attached figure using PETSc. The solution domain would be the shaded region only and it extends inside the cube. I would like to have suggestions on ways to form and assemble the matrix associated with the discretization. Also, is it a good practise to discretize the entire domain and blank the grid points which are within the cube not in the domain ?
 
Regards,
Dominic
 

Attachment: cube.jpg
Description: cube.jpg