I would like to solve for the electric potential and magnetic vector potential using the finite volume method (collocated grid). My equations are:$\nabla\cdot(\sigma\nabla\phi)=0$$\nabla \cdot \nabla A = \mu_m \sigma \nabla\phi$The domain is a wedge, with the following boundary conditions:A fixed gradient for the electric potential $\phi$ on the top, zero gradient at the bottom and a zero value at the side boundaryZero gradient for the magnetic vector potential $A$ on all boundariesI didn't expect this would be a tough problem, but the linear solver (GAMG) only converges with under-relaxation, and the result is dependent on the value of the under-relaxation factor. Can anyone give me a hint where things are going wrong?Additional information: The software package I use is OpenFOAM. The gradients are discretized using the Gauss linear scheme, and the laplacian with Gauss linear corrected (explicit non-orthogonal correction).
Finite volume solution of electrostatics using magnetic vector potential
convergence;finite volume
The solution required two changes:Change the boundary conditions for $A$ to fixedValue 0, which follows from $B\cdot n = 0$ at the boundaries.Disable under-relaxation. Even though the system converged without under-relaxation, I was not able to reach convergence no matter which value I assigned to the under-relaxation coefficient. This seems to be a problem with the implementation of matrix relaxation in OpenFOAM.