一类基于带约束能量最小基函数的数值均匀化方法的二维数值实现

doi:10.12288/szjs.s2021-0768

数值计算与计算机应用 ›› 2022, Vol. 43 ›› Issue (4): 363-379.DOI: 10.12288/szjs.s2021-0768

• 论文 • 上一篇

一类基于带约束能量最小基函数的数值均匀化方法的二维数值实现

刘新亮¹, 张镭¹, 朱圣鑫^2,3

1. 上海交通大学, 数学科学学院, 自然科学研究院, 教育部科学工程计算重点实验室, 上海 200240;
2. 北京师范大学数学研究中心, 珠海 519087;
3. 北京师范大学-香港浸会大学联合国际学院, 珠海 519087

收稿日期:2021-07-05 发布日期:2022-12-08
基金资助:
国家自然科学基金(11871339, 11861131004)资助.

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刘新亮, 张镭, 朱圣鑫. 一类基于带约束能量最小基函数的数值均匀化方法的二维数值实现[J]. 数值计算与计算机应用, 2022, 43(4): 363-379.

Liu Xinliang, Zhang Lei, Zhu Shengxin. NUMERICAL IMPLEMENTATION OF A CLASS OF NUMERICAL HOMOGENIZATION METHODS WITH BASES FROM CONSTRAINED ENERGY MINIMIZATION[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(4): 363-379.

NUMERICAL IMPLEMENTATION OF A CLASS OF NUMERICAL HOMOGENIZATION METHODS WITH BASES FROM CONSTRAINED ENERGY MINIMIZATION

Liu Xinliang¹, Zhang Lei¹, Zhu Shengxin^2,3

1. School of Mathematical Sciences, Institute of Natural Sciences, and MOE-LSC, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China;
2. Research Center for mathematics, Beijing Normal University, Zhuhai 519087, China;
3. Division of Science and Technology, BNU-HKBU United International College, Zhuhai 519087, China

Received:2021-07-05 Published:2022-12-08

近年来，多尺度偏微分方程的数值均匀化方法得到了快速发展.本文以Rough Polyharmonic Splines (RPS)及其推广形式Generalized Rough Polyharmonic Splines (GRPS)为例，介绍了一类基于带约束能量最小基函数的数值均匀化方法的数学形式，并详细给出了基于粗细两网格，且具有拟最优计算量和收敛性的局部化基函数的数值实现方法.我们对具有多尺度系数的二维椭圆方程验证了这类方法的收敛性,此类方法在简单修改后还可用于多尺度Helmholtz方程等其他问题.

关键词： 多尺度偏微分方程, 数值均匀化, 带约束能量最小化, 局部化基函数

Last decades have witnessed the fast development of numerical homogenization methods for multiscale PDEs. In this paper, we use Rough Polyharmonic Splines(RPS) and its generalization, namely Generalized Rough Polyharmonic Splines(GRPS) as representative examples, to introduce the mathematical formulation and numerical implementation of a class of numerical homogenization methods with bases from constrained energy minimization. We present details of the construction of two-level mesh, local patches, and the computation of localized bases with quasi-optimal computational complexity and accuracy. The methods are numerically justified for multiscale elliptic equations with rough coefficients, and they can be applied to other problems such as multiscale Helmholtz equations with minor modifications.

Key words: multiscale PDEs, numerical homogenization, constrained energy minimization, localized bases

MR(2010)主题分类:

65N22
65N50

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[[1] Owhadi H, Zhang L, Berlyand L. Polyharmonic homogenization, rough polyharmonic splines and sparse super-locatization[J]. ESIAM, 2014, 48(2):517-552.
[2] Liu X, Zhang L, S. Z. Generalized Polyharmonic Splines for Multiscale PDEs with Rough Coefficients[J]. Preprint, 2021.
[3] Målqvist A, Peterseim D. Localization of elliptic multiscale problems[J]. Math. Comp., 2014, 83(290):2583-2603.
[4] Chung E T, Efendiev Y, Leung W T. Constraint energy minimizing generalized multiscale finite element method[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 339:298- 319.
[5] Peterseim D. Eliminating the pollution effect in Helmholtz problems by local subscale correction.[J]. Math. Comp., 2017, 86:1005-1036.
[6] Liu X, Chung E, Zhang L. Iterated numerical homogenization for multi-scale elliptic equations with monotone nonlinearity[J]. arXiv preprint arXiv:2101.00818, 2021.
[7] Gräser C, Sander O. The dune-subgrid module and some applications[J]. Computing, 2009, 86(4):269.
[8] Alnæs M, Blechta J, Hake J, Johansson A, Kehlet B, Logg A, Richardson C, Ring J, Rognes M E, Wells G N. The FEniCS project version 1.5[J]. Archive of Numerical Software, 2015, 3(100).
[9] ZHANG L, ZHENG W, LU B, CUI T, LENG W, LIN D. The toolbox PHG and its applications[J]. SCIENTIA SINICA Informationis, 2016, 46(10):1442-1464.
[10] Owhadi H. Bayesian Numerical Homogenization[J]. Multiscale Model. Simul., 2015, 13(3):812- 828.
[11] Owhadi H, Zhang L. Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients[J]. accpted by Journal of Computational Physics, 2017.
[12] Owhadi H. Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games[J]. SIAM Rev., 2017, 59(1):99-149. arXiv:1503.03467, 2015.
[13] Braess D. Finite elements:Theory, fast solvers, and applications in solid mechanics[M]. Cambridge University Press, 2007.
[14] Boyd S, Boyd S P, Vandenberghe L. Convex optimization[M]. Cambridge university press, 2004.
[15] Li R, Chen Z, Wu W. Gneralized Difference Methods For Differential Equations[M]. Marcel Dekker, Inc, 2000.
[16] Zhang Z, Zou Q. Some recent advances on vertex centered finite volume element methods for elliptic equations[J]. Science China Mathematics, 2013, 56:2507-2522.

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摘要

一类基于带约束能量最小基函数的数值均匀化方法的二维数值实现

NUMERICAL IMPLEMENTATION OF A CLASS OF NUMERICAL HOMOGENIZATION METHODS WITH BASES FROM CONSTRAINED ENERGY MINIMIZATION

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