Stochastic Modeling and Regularity of the Nonlinear Elliptic Curl-Curl Equation

TL;DR

This paper investigates the stochastic nonlinear elliptic curl-curl equation relevant to magnetostatics, analyzing solution regularity and proposing numerical approximation methods under uncertainty in the B-H material law.

Contribution

It introduces a stochastic nonlinear curl-curl formulation with a truncated Karhunen-Loève approximation, and analyzes the solution's regularity and stochastic error decay.

Findings

Solution is not analytic with respect to random variables.

Algebraic decay of stochastic error observed.

Numerical results demonstrate the approximation approach.

Abstract

This paper addresses the nonlinear elliptic curl-curl equation with uncertainties in the material law. It is frequently employed in the numerical evaluation of magnetostatic fields, where the uncertainty is ascribed to the so-called B-H curve. A truncated Karhunen-Lo\`eve approximation of the stochastic B-H curve is presented and analyzed with regard to monotonicity constraints. A stochastic non-linear curl-curl formulation is introduced and numerically approximated by a finite element and collocation method in the deterministic and stochastic variable, respectively. The stochastic regularity is analyzed by a higher order sensitivity analysis. It is shown that, unlike to linear and several nonlinear elliptic problems, the solution is not analytic with respect to the random variables and an algebraic decay of the stochastic error is obtained. Numerical results for both the Karhunen-Lo\`eve expansion and the stochastic curl-curl equation are given for illustration.