含参双重连续介质模型的广义多尺度有限元迭代方法

马玲玲; 宋晓燕

doi:10.21656/1000-0887.460121

含参双重连续介质模型的广义多尺度有限元迭代方法

doi: 10.21656/1000-0887.460121

马玲玲^1,,
宋晓燕^2, ,

1.
湖南第一师范学院数学与统计学院湖南省大数据分析与应用重点实验室, 长沙 410205
2.
湖南工商大学数学与统计学院, 长沙 410205

基金项目:

国家自然科学基金 12101217

国家自然科学基金 12301551

国家自然科学基金 12331011

湖南省自然科学基金 2022JJ40113

湖南省自然科学基金 2022JJ40125

湖南省教育厅科学研究 22B0635

详细信息

作者简介:
马玲玲(1990—)，女，讲师，博士(E-mail: hudalingling@126.com)

通讯作者:
宋晓燕(1992—)，女，讲师，博士(通信作者. E-mail: xysonghnu@163.com)

中图分类号: O241.82; O331; TU45
计量
- 文章访问数: 36
- HTML全文浏览量: 14
- PDF下载量: 9
- 被引次数: 0
出版历程
- 收稿日期: 2025-06-11
- 修回日期: 2026-06-11
- 刊出日期: 2026-07-01

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

Ma Lingling^1
,,
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

摘要

摘要: 含参双重连续介质模型在地下地层相关的多种应用建模中具有重要的理论意义与实际价值. 该模型能够有效描述地质体中高度非均质性、强对比度的多尺度结构特征，并体现出显著的不确定性. 针对此类复杂模型的数值求解，采用合适的降阶方法已成为提升计算效率与保持求解精度的关键手段. 本文针对含参双重连续介质模型，提出了一种基于广义多尺度有限元方法的迭代求解策略. 该方法首先将原始的参数依赖型双重介质模型重新表述为一个包含多尺度扩散系数和转移函数(均与参数无关)以及参数相关右端项的新模型. 在此基础上，所提出的迭代方法划分为离线与在线两个阶段. 在离线阶段，本文在每个粗网格区域内，根据确定性多尺度参数构造多尺度基函数，并生成相应的降阶空间；在在线阶段，利用构建好的降阶空间，通过迭代方法对模型进行高效求解. 该方法的显著优势在于，离线阶段构造完成后，在线阶段每次迭代均可使用高效的直接求解器，并能重复利用矩阵逆，从而显著降低计算成本. 此外，本文还给出了该迭代方法的收敛性分析. 最后，通过参数依赖的双重连续介质模型的数值算例，验证了所提出方法的有效性与计算效率，并进一步验证了理论收敛结果的正确性.
- 双重连续介质模型 /
- 广义多尺度有限元方法 /
- 迭代方法 /
- 不确定性量化
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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图 1 网格剖分

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. The grid generation

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图 2 高震荡渗透率场κ₁⁰(x)(左)和κ₂⁰(x)(右)

Figure 2. High-contrast coefficients κ₁⁰(x) (left) and κ₂⁰(x) (right)

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图 3 不同数值模拟方法计算得到的均值轮廓图

Figure 3. Mean profiles obtained with different numerical simulation methods

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图 4 变量u(x; ξ)在200个随机样本处的相对误差

Figure 4. The relative errors of variable u(x; ξ) based on 200 random samples

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图 5 绝对误差随迭代次数变化的情况

Figure 5. Absolute errors of the iterations

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图 6 高对比函数

Figure 6. High-contrast functions

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图 7 双重连续问题的参考解和数值解

Figure 7. The reference solutions and numerical solutions of the dual-continuum problem

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表 1 不同粗网格大小H下，变量u(x; ξ)在L²范数意义下的相对误差

Table 1. The relative L² errors with different coarse mesh sizes H for u(x; ξ)

relative error	H=1/2	H=1/4	H=1/6	H=1/8	H=1/10	H=1/12	H=1/24
e_u^gms	6.996 8E-1	4.820 2E-1	3.099 6E-1	1.124 4E-1	2.844 9E-4	1.913 1E-4	1.208 5E-5
e_u^Gms-Iter	2.873 3E-1	2.243 0E-1	1.983 5E-1	1.615 3E-1	5.824 6E-3	6.213 6E-3	1.245 7E-3

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表 2 不同粗网格大小下，变量u(x; ξ)在能量范数下的相对误差

Table 2. The relative energy errors with different coarse mesh sizes H for u(x; ξ)

relative error	H=1/2	H=1/4	H=1/6	H=1/8	H=1/10	H=1/12	H=1/24
e_u^gms	2.492 0E-1	1.662 1E-1	1.415 8E-1	1.278 9E-1	6.701 0E-4	7.189 8E-4	4.979 2E-4
e_u^Gms-Iter	2.873 9E-1	2.244 5E-1	1.985 4E-1	1.617 6E-1	6.247 9E-3	6.636 7E-3	1.675 9E-3

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表 3 使用5个基函数的收敛性

Table 3. Convergences with 5 local basis functions

$\underline{\boldsymbol{\kappa}}_1^0(x)$	$\underline{\boldsymbol{\kappa}}_2^0(x)$	e_p^{1, Gms-Iter}	e_p^{2, Gms-Iter}	a_Q^Gms-Iter
1	1	1.747 0E101	3.970 0E105	1.644E107
3	3	3.670 3E52	9.801 2E57	4.323E59
10	10	2.409 0E-1	8.186 6E5	3.872 2E7
15	15	4.105 7E-3	1.338 5E-2	1.032E-2
20	20	2.822 0E-3	8.364 3E-3	8.832E-3

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表 4 基于L²范数意义的收敛结果

Table 4. Convergences based on the L²-norm

H	e_u^gms	order	e_u^Gms-Iter	order
1/2	6.079 4E-1	-	6.789 1	-
1/4	4.991 3E-1	0.284 5	5.535 8	0.294 4
1/8	2.991 3E-2	2.172 5	2.577 4E-2	4.020 6
1/16	2.366 1E-5	4.883 0	4.999 8E-3	3.469 0
1/32	4.960 3E-6	4.225 8	5.509 3E-3	2.566 8

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表 5 基于能量范数意义的收敛结果

Table 5. Convergences based on the energy norm

H	e_u^gms	order	e_u^Gms-Iter	order
1/2	6.648 8E-1	-	6.650 4E-1	-
1/4	5.821 0E-1	0.191 8	5.823 0E-1	0.191 7
1/8	5.419 5E-2	1.743 9	5.465 2E-2	3.478 4
1/16	3.546 3E-3	2.516 9	4.017 8E-3	2.457 0
1/32	1.680 7E-3	2.157 0	2.152 8E-3	2.067 8

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