We consider time-harmonic scalar transmission problems between dielectric and dispersive materials with generalized Lorentz frequency laws. For certain frequency ranges such equations involve a sign-change in their principle part. Due to the resulting loss of coercivity properties, the numerical simulation of such problems is demanding. Furthermore, the related eigenvalue problems are nonlinear and give rise to additional challenges. We present a new finite element method for both of these types of problems, which is based on a weakly coercive reformulation of the PDE. The new scheme can handle $C^{1,1}$-interfaces consisting piecewise of elementary geometries. Neglecting quadrature errors, the method allows for a straightforward convergence analysis. In our implementation we apply a simple, but nonstandard quadrature rule to achieve negligible quadrature errors. We present computational experiments in 2D and 3D for both source and eigenvalue problems which confirm the stability and convergence of the new scheme.

翻译：我们考虑电介质与具有广义洛伦兹频率规律的色散材料之间的时谐标量传输问题。在特定频率范围内，此类方程的主部会出现符号变化。由于由此导致的余强制性质丧失，这类问题的数值模拟具有挑战性。此外，相关的特征值问题是非线性的，带来了额外困难。我们针对这两类问题提出了一种基于PDE弱余强制重构的新有限元方法。该新方案能够处理由基本几何结构分段构成的$C^{1,1}$界面。忽略求积误差时，该方法允许进行直接的收敛性分析。在我们的实现中，采用了一种简单但非标准的求积规则以实现可忽略的求积误差。我们展示了二维和三维中源问题与特征值问题的计算实验，证实了新方案的稳定性与收敛性。