A Generalized Multiscale Iterative Finite Element Method for Parameterized Dual-Continuum Models

Ma Lingling; Song Xiaoyan

doi:10.21656/1000-0887.460121

Volume 47 Issue 7

Jul. 2026

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Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(7): 936-958

Ma Lingling, Song Xiaoyan. A Generalized Multiscale Iterative Finite Element Method for Parameterized Dual-Continuum Models[J]. Applied Mathematics and Mechanics, 2026, 47(7): 936-958. doi: 10.21656/1000-0887.460121

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A Generalized Multiscale Iterative Finite Element Method for Parameterized Dual-Continuum Models

doi: 10.21656/1000-0887.460121

Ma Lingling¹,
Song Xiaoyan^{2
,
,}

1. Hunan Provincial University Key Laboratory for Big Data Analysis and Application, School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, P.R. China;
2. School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha 410205, P.R. China

Funds:

The National Science Foundation of China(v)

Received Date: 2025-06-11
Rev Recd Date: 2026-06-11

Available Online: 2026-07-23

Abstract

Abstract

The parameterized dual-continuum model plays a significant theoretical and practical role in various subsurface geological modeling applications. This model effectively captures the highly heterogeneous, high-contrast, and multiscale structural features of geological formations, while also exhibiting considerable uncertainty. To address the numerical challenges posed by such complex models, adopting appropriate model reduction techniques has become a key approach to improving computational efficiency while maintaining solution accuracy. Herein an iterative solution strategy was proposed based on the uncoupled generalized multiscale finite element method (GMsFEM) for the parameterized dual-continuum model. The original parameter-dependent model was reformulated as a new one consisting of multiscale diffusion coefficients and transfer functions (both parameter-independent), along with a parameter-dependent source term. The proposed iterative method was divided into offline and online stages. In the offline stage, multiscale basis functions were constructed in each coarse grid block based on deterministic multiscale parameters to generate a reduced-order space. In the online stage, the model was solved efficiently within the reduced space with an iterative scheme. A major advantage of this method lies in the fact that, once the multiscale basis space is constructed offline, each online iteration can leverage efficient direct solvers and reuse matrix inverses, thus significantly reducing computational costs. Furthermore, the convergence of the proposed iterative method was analyzed. Finally, numerical experiments on the parameterized dual-continuum model were conducted to demonstrate the effectiveness and efficiency of the multiscale-based iterative approach, and validate the theoretical convergence results.
- dual continuum model,
- generalized multiscale finite element method,
- iterative method,
- uncertainty quantification

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