We propose a multiscale method for mixed-dimensional elliptic problems with highly heterogeneous coefficients arising, for example, in the modeling of fractured porous media. The method is based on the Localized Orthogonal Decomposition (LOD) framework and constructs locally supported, problem-adapted basis functions on a coarse mesh that does not need to resolve the coefficient oscillations. These basis functions are obtained in parallel by solving localized fine-scale problems. Our a priori error analysis shows that the method achieves optimal convergence with respect to the coarse mesh size, independent of the coefficient regularity, with an exponentially decaying localization error. Numerical experiments validate these theoretical findings and demonstrate the computational viability of the method.

翻译：本文针对混合维椭圆问题提出了一种多尺度方法，该方法适用于高异质系数情形，例如在裂隙多孔介质建模中出现的问题。该方法基于局部正交分解框架，在无需解析系数振荡的粗网格上构造具有局部支撑、适应问题特性的基函数。这些基函数通过并行求解局部细尺度问题获得。我们的先验误差分析表明，该方法在粗网格尺寸上达到最优收敛，且收敛性与系数正则性无关，局部化误差呈指数衰减。数值实验验证了这些理论结果，并证明了该方法的计算可行性。