Partial differential equations are frequently solved using a global basis, such as the Fourier series, due to excellent convergence. However, convergence becomes impaired when discontinuities are present due to the Gibbs phenomenon, negatively impacting simulation speed and possibly generating spurious solutions. We resolve this by supplementing the smooth global basis with an inherently discontinuous basis, incorporating knowledge of the location of the discontinuities. The solution's discontinuities are reproduced with exponential convergence, expediting simulations. The highly constrained discontinuous basis also eliminates the freedom to generate spurious solutions. We employ the combined smooth and discontinuous bases to construct a solver for the modes of a resonator in an open electromagnetic system. These modes can then expand any scattering problem for any source configuration or incidence condition without further numerics, enabling ready access and physical insight into the spatial variation of Green's tensor. Solving for the modes is the most numerically intensive and difficult step of modal expansion methods, so our mode solver overcomes the last major impediment to the use of modal expansion for open systems.

由于极好的收敛性，经常使用全局基础（例如，傅里叶级数）来求解偏微分方程。但是，当由于吉布斯现象而存在不连续性时，会削弱收敛性，从而对仿真速度产生负面影响，并可能生成虚假解。我们通过使用固有的不连续性基础补充平滑的全局基础，并结合不连续性位置的知识来解决此问题。该解决方案的不连续性通过指数收敛再现，从而加快了仿真速度。高度受限的不连续基础还消除了生成伪解的自由。我们采用组合的光滑和不连续基来构造开放电磁系统中谐振器模式的求解器。然后，这些模式可以扩展任何源配置或入射条件下的任何散射问题，而无需进一步的数值，从而可以立即访问和物理洞悉格林张量的空间变化。求解模态是模态展开方法中最耗费数值和最困难的步骤，因此我们的模态求解器克服了在开放系统中使用模态展开的最后一个主要障碍。