Convergence of solutions for a network approximation of the two-dimensional Laplace equation in a domain with a system of absolutely conducting disks

A boundary value problem for the Laplace equation describing the (electric, thermal, etc.) field of a system of ideally conducting disks of radius $R$ is considered. The solution to the problem is analyzed under the condition that the characteristic distance $\delta$ between the disks is small. It was previously proved that the original continuous problem can be approximated as $\delta\to0$ by a finite-dimensional network problem in the sense that the effective conductivities (energies) of the continuous problem are close to those of its network model. It is shown that the potentials of the ideally conducting disks determined from the continuous problem and the network model are also close to each other as $\delta\to0$, and the difference between the potentials is $O(\varepsilon^{1/4})$, where $\varepsilon=\delta/R$ is the characteristic relative distance between the disks.