This contribution extends the localized training approach, traditionally employed for multiscale problems and parameterized partial differential equations (PDEs) featuring locally heterogeneous coefficients, to the class of linear, positive symmetric operators, known as Friedrichs' operators. Considering a local subdomain with corresponding oversampling domain we prove the compactness of the transfer operator which maps boundary data to solutions on the interior domain. While a Caccioppoli-inequality quantifying the energy decay to the interior holds true for all Friedrichs' systems, showing a compactness result for the graph-spaces hosting the solution is additionally necessary. We discuss the mixed formulation of a convection-diffusion-reaction problem where the necessary compactness result is obtained by the Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario involving heterogeneous diffusion fields with multiple high-conductivity channels, demonstrate the effectiveness of the proposed method.

翻译：本文拓展了传统上用于多尺度问题及含局部异质系数参数化偏微分方程（PDE）的局部化训练方法，将其应用于线性正定对称算子类（即弗里德里希算子）。考虑带相应过采样域的局部子域，我们证明了将边界数据映射至内部区域解空间的传递算子的紧致性。尽管刻画内部能量衰减的卡乔波利不等式对所有弗里德里希系统均成立，但还需证明承载解空间的图空间的紧性结果。针对对流-扩散-反应问题的混合形式，我们通过皮卡德-韦克-韦伯定理获得了必要的紧性结果。聚焦于含多条高导通道的异质扩散场的数值实验证明了所提方法的有效性。