Hi All:

This is really a question in harmonic analysis - Let me state the question and then the CFD setting.

I am solving the Poisson equation in 2D

Lap(u) = f(x,y)

in a 2d doamin. There are both Dirchlet and Neumann conditions on the boundary. My question Is:

1) To what extent does the solution u(x,y) depend on values of f(x,y) ON the boundary ?

My guess is that on segments where there is a Dirichlet condition, values of f on the boundary do not affect the solution at all. But what about on other boundary segments ?

The setting of this is when you are solving for the stream function from the vorticity. Typically you know f(x,y) only on interior points and do not know it on the boundary.

Thanks !

Suresh

If the Poisson equation is valid throughout the domain (including the boundary), the Laplacian of u is given by f everywhere, including the boundary. That means, even though u itself (Dirichlet) or its gradient (Neumann) are specified on the boundary, the Laplacian there still depends on f, and that should have some effect on the interior solution.

This is an interesting issue. What I want to emphasize is that the equation is not valid at the boundary but valid in the neighborhood of the boundary, no matter of the BC being adopted.

The equation must be valid at the boundary if the solution of the boundary value problem is to be solved. It's just a matter of how you interpret the limiting process as the boundary is approached; i.e. the derivatives are one sided at the boundary.

Possibly more important for your question is the compatability relation which tells you whether the problem is solvable; e.g. integrate your equation over the whole domain and apply the divergence theorem.