# linear elliptic vector equation

nicolas aunai nicolas.aunai at gmail.com
Tue Aug 18 02:30:38 CDT 2009

```Dear all,

I have a simulation code (particle in cell) in which I have to solve a
linear vector equation 4 times per time step. A typical run consists
of 50 000 time steps. The grid is rectangular and uniform of size Lx,
Ly, with nx+1 and ny+1 points in the x and y direction respectively.,
(nx, ny) could be at max (1024,1024).

The vector equation is the following :

B(x,y) - a*Laplacian(B(x,y)) = S(x,y)

Where B and S are 2D vector fields with 3 components (Bx, By, Bz and
Sx, Sy, Sz), each depending on the x and y coordinates.

'a' is a positive constant, smaller than one, typically a= 0.02

Boundary conditions may depend on for which B component we are solving
the equation. On the x=cst borders, the boundary is always periodic,
no matter what component is solved, but on y=cst borders, the boundary
condition can be either Neumann or Dirichlet.

So far, I have written a small code that creates a vector solution, a
vector RHS and the matrix operator, and solve a scalar equation of
this type. Solving my vector equation would then just call 3 times
this kind of code... but I believe there is another way to deal with
vector fields and linear systems with Petsc, using DAs no ? Could
someone explain me how solve this the proper way and/or show me some
code solving linear system with vector fields ? (is there an example
in the exercices that I would not have seen ?)

Thanks a lot
Nico
```