An Online Generalized Multiscale Finite Element Method for Dual-continuum Unsaturated Filtration Problem in Domains with Rough Boundaries

The paper explores the dual-continuum unsaturated filtration model in heterogeneous porous media with rough surface topography. The rough surface topography plays a crucial role in determining the flow process and incorporates multiscale features. The mathematical model is based on the Richard’s equation with a Robin boundary condition for the rough surface boundary. The Online Generalized Multiscale Finite Element Method (Online GMsFEM) is used for coarse-grid discretization. Multiscale basis functions that encompass small-scale heterogeneities are created. Additional basis functions are generated to accommodate the impact of boundary conditions on rough surfaces. We present a consrtuction of online mulsticale basis functions that based on local residuals. All multiscale basis functions are coupled and take into account transfer term between continuums. Numerical results for a two-dimensional problem are provided, and relative errors between multiscale and reference solutions are calculated for varying numbers of multiscale basis functions to verify the results. The influence of online bases on the algorithm’s accuracy is also discussed.