[petsc-users] Question about periodic conditions

[Patrick Sanan]

Hi Zakariae - sorry about the delay - responses inline below.

I'd be curious to see your code (which you can send directly to me if you don't want to post it publicly), so I can give you more comments, as DMStag is a new component. 


> Am 23.03.2021 um 00:54 schrieb Jorti, Zakariae <zjorti at lanl.gov>:
> 
> Hi, 
> 
> I implemented a PETSc code to solve Maxwell's equations for the magnetic and electric fields (B and E) in a cylinder: 
> 0 < r_min <= r <= r_max;                with r_max > r_min
> phi_min = 0 <= r <= phi_max  = 2 π
> z_min <= z =< z_max;                    with z_max > z_min.
> 
> I am using a PETSc staggered grid with the electric field E defined on edge centers and the magnetic field B defined on face centers. (dof0 = 0, dof1 = 1,dof2 = 1, dof3 = 0;).  
> 
> I have two versions of my code: 
> 1 - A first version in which I set the boundary type to DM_BOUNDARY_NONE in the three directions r, phi and z
> 2- A second version in which I set the boundary type to DM_BOUNDARY_NONE in the r and z directions, and DM_BOUNDARY_PERIODIC in the phi direction. 
> 
> When I print the solution vector X, which contains both E and B components, I notice that the vector is shorter with the second version compared to the first one. 
> Is it normal?  
Yes - with the periodic boundary conditions, there will be fewer points since there won't be the "extra" layer of faces and edges at phi  = 2 * pi .

If you consider a 1-d example with 1 dof on vertices and cells, with three elements, the periodic case looks like this, globally,

    x ---- x ---- x ----

as opposed to the non-periodic case,

   x ---- x ---- x ---- x


> 
> Besides, I was wondering if I have to change the way I define the value of the solution on the boundary. What I am doing so far in both versions is something like:
> B_phi [phi = 0] = 1.0;
> B_phi [phi = 2π] = 1.0;
> E_z [r, phi = 0] = 1/r;
> E_z [r, phi = 2π] = 1/r;
> 
> Assuming that values at phi = 0 should be the same as at phi=2π with the periodic boundary conditions, is it sufficient for example to have only the following boundary conditions:
> B_phi [phi = 0] = 1.0;
> E_z [r, phi = 0] = 1/r ? 

Yes - this is the intention, since the boundary at phi = 2 * pi is represented by the same entries in the global vector. 

Of course, you need to make sure that your continuous problem is well-posed, which in general could change when using different boundary conditions. 

> Thank you.
> Best regards,
> 
> Zakariae Jorti

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