A Localized Orthogonal Decomposition Method for Heterogeneous Mixed-Dimensional Problems
This work addresses computational difficulties in modeling fractured porous media for applications like hydrology or geophysics, but it appears incremental as it builds on the existing LOD framework.
The authors tackled the challenge of solving mixed-dimensional elliptic problems with highly heterogeneous coefficients, such as those in fractured porous media modeling, by proposing a multiscale method based on the Localized Orthogonal Decomposition framework, which achieves optimal convergence independent of coefficient regularity and shows exponential decay in localization error in numerical experiments.
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.