# Iterative solver, which balances the equations with the biggest errors

I am trying to understand the mathematical side of an algorithm, which approximates the solution of coulomb's law. Their approach is to balance the two elements with the biggest positive/negative error.

Trying to understand the stability conditions of the algorithm, I am searching if there is an iterative solver with an similiar approach, as the problem can be stated as a linear system (but has a lot more computational complexity when solving the system with usual solvers).

The algorithm does not use any derivatives, but reduces the error induced from two two elements with the biggest residuum on each other to zero and recalculates the residuum of all other elements after the balancing.

The potential they are balancing is defined by $U_i = \sum_j I_{ij} q_j$ at points $x_i$ at the center of triangles $\Delta S_j$ with charge $q_j$, where $$I_{ij} = \frac{1}{\Delta S_j} \int_{\Delta S_j}k|\vec{x}_i-\vec{x}'|^{-1}dS'.$$

In each step the points m, n with the biggest positive/negative deviation to an equipotential are balanced by transferting a charge $$q=\frac{U_m - U_n}{I_{mm}+I_{nn}-I_{mn}-I_{nm}}$$ giving the new potentials at the points $x_i, x_j$:

$$U'_m = U_m - I_{mm}q + I_{mn}q \\ U'_n = U_n + I_{mm}q - I_{mn}q$$

After this transfer $U'_m = U'_n$ holds and all other potentials $U_i$ are updated:

$$U'_i = U_i + q_m(U'_m - U_m) + q_m(U'_n - U_n)$$

After each step, the potentials $U'_n, U'_m$ are equal and in the setting of electrostatics all other potentials come closer to an equipotential on the surface as well.

I suspect, that this way of optimization is already described somewhere else with more numerical analysis about errors, convergence criteria and stability conditions, but was not able to find it in literature, yet.

• You're unlikely going to get a lot of responses is your question requires reading a 40-page paper. Can you try to make your question self-contained, including using formulas where necessary? – Wolfgang Bangerth Aug 6 '17 at 22:04
• The essence of the question is the second sentence: I am searching a solver, which balances the two biggest errors (positive/negative) in each iteration. The paper is quoted for completedness. They define the procedure as an algorithm, but I suspect it is equivalent to a known numeric solver/optimization algorithm. – allo Aug 7 '17 at 5:37