Hierarchical multiscale finite element method for multi-continuum media

Abstract

Simulation in media with multiple continua where each continuum interacts with every other is often challenging due to multiple scales and high contrast. One needs some type of model reduction. One of the approaches is a multi-continuum technique, where every process in each continuum is modeled separately and an interaction term is added. Direct numerical simulation in multiscale media is usually not practicable. For this reason, one constructs the corresponding homogenized equations which approximate the solutions to the multiscale equations when the microscopic scales tend to 0. Computing the effective coefficients of the homogenized equations can be expensive because one needs to solve local cell problems for a large number of macroscopic points. The paper considers a two scale two continuum system where the interaction terms between the continua are scaled as O(1∕ϵ2) where ϵ is the microscopic scale. We prove that in the homogenization limit, we obtain the same limit for both continua. We derive the homogenized equation for the limit function; and prove the homogenization convergence rigorously. The homogenized coefficients are established from solutions of the cell problems which are systems of equations of a similar form as the two continuum system. We develop a hierarchical approach for solving these cell problems at a dense network of macroscopic points with an essentially optimal computation cost. The method employs the fact that neighboring representative volume elements (RVEs) share similar features; and effective properties of the neighboring RVEs are close to each other. The hierarchical approach reduces computation cost by using different levels of resolution for cell problems at different macroscopic points. Solutions of the cell problems which are solved with a higher level of resolution are employed to correct the solutions at neighboring macroscopic points that are computed by approximation spaces with a lower level of resolution. The method requires a hierarchy of macroscopic grid points and corresponding nested approximation spaces with different levels of resolution. Each level of macroscopic points is assigned to an approximation finite element (FE) space which is used to solve the cell problems at the macroscopic points from that level. We prove rigorously that this hierarchical method achieves the same level of accuracy as that of the full solve where cell problems at every macroscopic point are solved using FE spaces with the highest level of resolution, but at the essentially optimal computation cost. Numerical implementation that computes effective permeabilities of a two-scale multicontinuum system via the numerical solutions of the cell problems supports the analytical results.