# Can Gauss-Seidel/SOR (preconditioned?) be applied to a non-diagonally dominant matrix?

After applying finite difference method to a Laplace/Poisson problem always arises a diagonal dominant system of equations that can be solved with Gauss-Seidel or SOR methods. If the original PDE does not have a second order derivative (Laplace operator), but a first order operator (divergence or curl), the arising system of equations is not diagonal dominant, thus we need to solve it with another solver.

There any way to change it (pivoting, preconditioning, ...) so that it can be solved with Gauss-Seidel or SOR methods?

• I think the question is missing something: the equation $\nabla\times u=f$ doesn't have a solution unless $\nabla\cdot f=0$ (similar to the "boundary problem" $u'=f$, $u(a)=u(b)$). So before you apply GS or SOR to such a discretized system (what is the equation exactly?), you must ask if the equation is well-posed, and you must make sure the system is nonsingular, etc. A lot of these methods were developed with diagonally dominant matrices in mind. – Kirill Mar 21 '15 at 20:43
• It isn't a question about an specific problem, but consider that I'm trying to solve magnetostatic field with $\nabla \times\vec{H}=\vec{J}_0$, $\nabla \cdot\vec{B}=0$ and $\vec{B} = \mu_0\vec{H}$. How can I discretize $\nabla \times\vec{H}=\vec{J}_0$ with finite central difference of second order or more in a way that GS or SOR will be applicable? – user3368561 Mar 21 '15 at 20:55
• In other words. I have this discretized equation $\frac{\partial u}{\partial x} \approx \frac{u_{i+1}-u_{i-1}}{\Delta x} = f$. How to transform it to allow solve it with GS or SOR? – user3368561 Mar 21 '15 at 21:46
• Think about how the information is flowing in determining your solution. In an elliptic problem, boundary value information flow to the interior of your domain and discretely this corresponds to solving a system of simultaneous equations. In the discretized equation shown above the information would seem to flow naturally in one direction or the other (increasing $x$ or decreasing). It doesn't seem a compelling opportunity to rewrite the solution procedure/discretization to make it fit with solving a simultaneous system, as this would normally be more costly. – hardmath Mar 21 '15 at 21:57
• May be true, but given some boundary conditions, there is one solution of $u_i$ that holds true previous equation. I want to find those values using GS or SOR. How? – user3368561 Mar 22 '15 at 11:44