A finite element method for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model consists of a nonlinear Helmholtz equation that is reduced to a spherical domain. The (exemplary) finite element method is formed by Courant-type elements with curved facets at the boundary of the spherical computational domain. This method is examined for its well-posedness, in particular the validity of a discrete inf-sup condition of the modified sesquilinear form uniformly with respect to both the truncation and the mesh parameters is shown. Under suitable assumptions to the nonlinearities, a quasi-optimal error estimate is obtained. Finally, the satisfiability of the approximation property of the finite element space required for the solvability of a class of adjoint linear problems is discussed.

翻译：提出并研究了一种有限元方法，用于逼近可穿透有界物体（障碍物）对外部电磁场激励响应的数学模型的解。该模型由简化为球域的非线性亥姆霍兹方程构成。有限元方法（示例性）采用库朗型单元，并在球计算域边界处配置曲面面片。本文检验了该方法的适定性，特别证明了修正半双线性形式的离散inf-sup条件对截断参数和网格参数的一致有效性。在非线性项的适当假设下，获得了拟最优误差估计。最后，讨论了为求解一类伴随线性问题所需的有限元空间逼近性质的可满足性。