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This paper addresses the nonlinear elliptic curl-curl equation withuncertainties in the material law. It is frequently employed in the numericalevaluation of magnetostatic fields, where the uncertainty is ascribed to theso-called B-H curve. A truncated Karhunen-Lo\`eve approximation of thestochastic B-H curve is presented and analyzed with regard to monotonicityconstraints. A stochastic non-linear curl-curl formulation is introduced andnumerically approximated by a finite element and collocation method in thedeterministic and stochastic variable, respectively. The stochastic regularityis analyzed by a higher order sensitivity analysis. It is shown that, unlike tolinear and several nonlinear elliptic problems, the solution is not analyticwith respect to the random variables and an algebraic decay of the stochasticerror is obtained. Numerical results for both the Karhunen-Lo\`eve expansionand the stochastic curl-curl equation are given for illustration.

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