# Hierarchical multiscale finite element method for multi-continuum media

## TL;DR

This paper introduces a hierarchical multiscale finite element method for efficiently simulating multi-continuum media with multiple scales and high contrast, achieving optimal computational cost while maintaining accuracy.

## Contribution

It develops a hierarchical approach that reduces computation cost by using different resolution levels for cell problems across a dense network of macroscopic points.

## Key findings

- The method achieves the same accuracy as full high-resolution solves.
- Numerical results validate the analytical convergence and efficiency.
- The approach effectively computes effective permeabilities in multi-continuum systems.

## 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 types of model reduction. One of the approaches is multi-continuum technique, where every process in each continuum is modeled separately and an interaction term is added. Direct numerical simulation in multi scale media is usually not practicable. For this reason, one constructs the corresponding homogenized equations. The paper develops a hierarchical approach for solving 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. The method requires a hierarchy of macroscopic grid points and a 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 in 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 the 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. Finally, we prove the homogenization convergence for our multiscale multi-continuum system.