This work studies the parameter-dependent diffusion equation in a two-dimensional domain consisting of locally mirror symmetric layers. It is assumed that the diffusion coefficient is a constant in each layer. The goal is to find approximate parameter-to-solution maps that have a small number of terms. It is shown that in the case of two layers one can find a solution formula consisting of three terms with explicit dependencies on the diffusion coefficient. The formula is based on decomposing the solution into orthogonal parts related to both of the layers and the interface between them. This formula is then expanded to an approximate one for the multi-layer case. We give an analytical formula for square layers and use the finite element formulation for more general layers. The results are illustrated with numerical examples and have applications for reduced basis methods by analyzing the Kolmogorov n-width.

翻译：本研究探讨了由局部镜像对称层构成的二维区域中的参数依赖扩散方程。假设扩散系数在各层内为常数。目标是构建具有少量项的近似参数-解映射。研究表明，在双层情况下，可得到由三项组成的解公式，其中各项对扩散系数具有显式依赖关系。该公式基于将解分解为与两层及界面相关的正交分量。随后将此公式推广至多层情形的近似解。针对方形层给出了解析公式，并对更一般的层采用有限元公式进行求解。通过数值算例验证了结果，并通过分析Kolmogorov n-宽度展示了该方法在降基法中的应用价值。