The Behavior of the Direct Current Density at the Edge of Electrodes

SIAM Journal on Applied Mathematics, Volume 86, Issue 2, Page 596-614, April 2026. Abstract. We study the shunt boundary condition for second order elliptic PDEs. A specific case of this is the PDE describing the electrostatic potential for a conductive body into which current is injected through electrodes that touch the boundary. We obtain the optimal description of the gradient of the electrostatic potential upon approach to the edge of the electrodes. This generalizes the classical Zaremba problem. The asymptotic description we obtain is then implemented to produce a substantial speed-up of numerical solvers for this boundary-value problem.