Efficient Jacobian Matrix Determination for $H^{2}$ Representations of Nonlinear Electrostatic Surface Integral Equations

In this paper, a nonlinear electrostatic surface integral equation is presented that is suitable for predicting corrosion-related fields. Nonlinear behavior arises due to electrochemical reactions at polarized surfaces. Hierarchical
$H^{2}$
matrices are used to compress the discretized integral equation for the fast solution of large problems. A technique based on randomized linear algebra is discussed for the efficient computation of the Jacobian matrix required at each iteration of a nonlinear solution.