Higher-Order Multiscale Computational Method for Multi-Continuum Problems in Highly Heterogeneous Media

TL;DR

This paper introduces a high-order multiscale computational method for accurately solving multi-continuum problems in highly heterogeneous media, combining homogenization, FEM, and interpolation techniques.

Contribution

The paper develops a higher-order multiscale asymptotic solution and a numerical algorithm that improves accuracy and efficiency for complex heterogeneous media problems.

Findings

The HOMS method achieves high accuracy in heterogeneous media simulations.

The numerical algorithm demonstrates stability and efficiency in experiments.

Convergence rate of the solution is rigorously established.

Abstract

This paper presents a high-accuracy higher-order multiscale method for solving multi-continuum problems in in highly heterogeneous media. First, microscopic unit cell functions are defined, leading to the derivation of macroscopic homogenized equations and formulas for calculating effective parameters, which yield a higher-order multi-scale (HOMS) asymptotic solution. Subsequently, the pointwise approximation properties of this solution to the original equations are analyzed, and its convergence rate in the integral norm is rigorously established under certain assumptions. Furthermore, a multiscale numerical algorithm is developed by integrating the finite element method (FEM), finite difference method, and interpolation technique. Finally, numerical experiments demonstrate the high accuracy, efficiency, and stability of the proposed HOMS numerical algorithm.