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

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--Loève approximation of the stochastic $B$--$H$ curve is presented and analyzed with regard to monotonicity constraints. A stochastic nonlinear curl--curl formulation is introduced and numerically approximated by a finite element and collocation method in the deterministic and the stochastic variable, respectively. The stochastic regularity is analyzed by a higher-order sensitivity analysis. It is shown that, unlike in 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--Loève expansion and the stochastic curl--curl equation are given for illustration.

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