Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some convergence analysis for multicontinuum homogenization

Wing Tat Leung

TL;DR

This work develops a multicontinuum homogenization framework for high-contrast multiscale flow without requiring scale separation. It combines Constraint Energy Minimizing Multiscale Finite Element Method (CEM-GMsFEM) with nonlocal multicontinuum (NLMC) concepts to build downscaling operators and a z-averaged homogenized operator that preserves microstructural effects in a macroscopic PDE with coefficients derived from local basis functions $\eta_{x,i}$ and their linear variants. The authors establish residual-based error estimates and, under a bounded inverse assumption for the effective operator, prove convergence of the upscaled solution to the true solution, while offering practical numerical strategies (NLMC-based coefficients and RVE oversampling) to reduce computation. The framework is designed to be discretization-agnostic and applicable to scenarios with multiple continua and high contrast, potentially enabling efficient and accurate simulations in complex porous media.

Abstract

In this paper, we provide an analysis of a recently proposed multicontinuum homogenization technique. The analysis differs from those used in classical homogenization methods for several reasons. First, the cell problems in multicontinuum homogenization use constraint problems and can not be directly substituted into the differential operator. Secondly, the problem contains high contrast that remains in the homogenized problem. The homogenized problem averages the microstructure while containing the small parameter. In this analysis, we first based on our previous techniques, CEM-GMsFEM, to define a CEM-downscaling operator that maps the multicontinuum quantities to an approximated microscopic solution. Following the regularity assumption of the multicontinuum quantities, we construct a downscaling operator and the homogenized multicontinuum equations using the information of linear approximation of the multicontinuum quantities. The error analysis is given by the residual estimate of the homogenized equations and the well-posedness assumption of the homogenized equations.

Some convergence analysis for multicontinuum homogenization

TL;DR

This work develops a multicontinuum homogenization framework for high-contrast multiscale flow without requiring scale separation. It combines Constraint Energy Minimizing Multiscale Finite Element Method (CEM-GMsFEM) with nonlocal multicontinuum (NLMC) concepts to build downscaling operators and a z-averaged homogenized operator that preserves microstructural effects in a macroscopic PDE with coefficients derived from local basis functions and their linear variants. The authors establish residual-based error estimates and, under a bounded inverse assumption for the effective operator, prove convergence of the upscaled solution to the true solution, while offering practical numerical strategies (NLMC-based coefficients and RVE oversampling) to reduce computation. The framework is designed to be discretization-agnostic and applicable to scenarios with multiple continua and high contrast, potentially enabling efficient and accurate simulations in complex porous media.

Abstract

In this paper, we provide an analysis of a recently proposed multicontinuum homogenization technique. The analysis differs from those used in classical homogenization methods for several reasons. First, the cell problems in multicontinuum homogenization use constraint problems and can not be directly substituted into the differential operator. Secondly, the problem contains high contrast that remains in the homogenized problem. The homogenized problem averages the microstructure while containing the small parameter. In this analysis, we first based on our previous techniques, CEM-GMsFEM, to define a CEM-downscaling operator that maps the multicontinuum quantities to an approximated microscopic solution. Following the regularity assumption of the multicontinuum quantities, we construct a downscaling operator and the homogenized multicontinuum equations using the information of linear approximation of the multicontinuum quantities. The error analysis is given by the residual estimate of the homogenized equations and the well-posedness assumption of the homogenized equations.
Paper Structure (7 sections, 4 theorems, 64 equations)

This paper contains 7 sections, 4 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multicontinumm homogenization
  4. Analysis
  5. Numerical Approaches
  6. Conclusion
  7. Acknowledgement

Key Result

Theorem 1

Let $f\in L^{2}(\Omega)$ and $u_{glo,H_{\epsilon}}^{z}\in V_{glo,H_{\epsilon}}^{z}$ be the solution of We have and

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof