Journal articles: 'Multiscale problems'

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Cohen, Albert, Wolfgang Dahmen, Ronald DeVore, and Angela Kunoth. "Multiscale and High-Dimensional Problems." Oberwolfach Reports 10, no. 3 (2013): 2179–257. http://dx.doi.org/10.4171/owr/2013/39.

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Cohen, Albert, Wolfgang Dahmen, Ronald DeVore, and Angela Kunoth. "Multiscale and High-Dimensional Problems." Oberwolfach Reports 14, no. 1 (January 2, 2018): 1001–51. http://dx.doi.org/10.4171/owr/2017/17.

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WIJESEKERA, NIMAL, GUOGANG FENG, and THOMAS L. BECK. "MULTISCALE ALGORITHMS FOR EIGENVALUE PROBLEMS." Journal of Theoretical and Computational Chemistry 02, no. 04 (December 2003): 553–61. http://dx.doi.org/10.1142/s0219633603000665.

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Iterative multiscale methods for electronic structure calculations offer several advantages for large-scale problems. Here we examine a nonlinear full approximation scheme (FAS) multigrid method for solving fixed potential and self-consistent eigenvalue problems. In principle, the expensive orthogonalization and Ritz projection operations can be moved to coarse levels, thus substantially reducing the overall computational expense. Results of the nonlinear multiscale approach are presented for simple fixed potential problems and for self-consistent pseudopotential calculations on large molecules. It is shown that, while excellent efficiencies can be obtained for problems with small numbers of states or well-defined eigenvalue cluster structure, the algorithm in its original form stalls for large-molecule problems with tens of occupied levels. Work is in progress to attempt to alleviate those difficulties.

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Hyman, J. M. "Patch Dynamics for Multiscale Problems." Computing in Science and Engineering 7, no. 3 (May 2005): 47–53. http://dx.doi.org/10.1109/mcse.2005.57.

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Målqvist, Axel. "Multiscale Methods for Elliptic Problems." Multiscale Modeling & Simulation 9, no. 3 (July 2011): 1064–86. http://dx.doi.org/10.1137/090775592.

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Proksch, Katharina, Frank Werner, and Axel Munk. "Multiscale scanning in inverse problems." Annals of Statistics 46, no. 6B (December 2018): 3569–602. http://dx.doi.org/10.1214/17-aos1669.

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Målqvist, Axel, and Daniel Peterseim. "Localization of elliptic multiscale problems." Mathematics of Computation 83, no. 290 (June 16, 2014): 2583–603. http://dx.doi.org/10.1090/s0025-5718-2014-02868-8.

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Kickinger, Ferdinand. "Multiscale Problems; Meshes and Solvers." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 78, S3 (1998): 963–64. http://dx.doi.org/10.1002/zamm.19980781552.

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Masud, Arif, and Leopoldo P. Franca. "A hierarchical multiscale framework for problems with multiscale source terms." Computer Methods in Applied Mechanics and Engineering 197, no. 33-40 (June 2008): 2692–700. http://dx.doi.org/10.1016/j.cma.2007.12.024.

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Song, Fei, and Weibing Deng. "Multiscale discontinuous Petrov-Galerkin method for the multiscale elliptic problems." Numerical Methods for Partial Differential Equations 34, no. 1 (August 16, 2017): 184–210. http://dx.doi.org/10.1002/num.22191.

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Burov, Dmitry, Dimitrios Giannakis, Krithika Manohar, and Andrew Stuart. "Kernel Analog Forecasting: Multiscale Test Problems." Multiscale Modeling & Simulation 19, no. 2 (January 2021): 1011–40. http://dx.doi.org/10.1137/20m1338289.

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Hellman, Fredrik, and Axel Målqvist. "Contrast Independent Localization of Multiscale Problems." Multiscale Modeling & Simulation 15, no. 4 (January 2017): 1325–55. http://dx.doi.org/10.1137/16m1100460.

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Ming, Pingbing, and Xingye Yue. "Numerical methods for multiscale elliptic problems." Journal of Computational Physics 214, no. 1 (May 2006): 421–45. http://dx.doi.org/10.1016/j.jcp.2005.09.024.

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Frederick, Christina, and Björn Engquist. "Numerical methods for multiscale inverse problems." Communications in Mathematical Sciences 15, no. 2 (2017): 305–28. http://dx.doi.org/10.4310/cms.2017.v15.n2.a2.

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Heida, Martin, Ralf Kornhuber, and Joscha Podlesny. "Fractal Homogenization of Multiscale Interface Problems." Multiscale Modeling & Simulation 18, no. 1 (January 2020): 294–314. http://dx.doi.org/10.1137/18m1204759.

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Chen, Zhangxin. "Multiscale methods for elliptic homogenization problems." Numerical Methods for Partial Differential Equations 22, no. 2 (2006): 317–60. http://dx.doi.org/10.1002/num.20099.

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Guo, Ning, and Jidong Zhao. "Multiscale insights into classical geomechanics problems." International Journal for Numerical and Analytical Methods in Geomechanics 40, no. 3 (July 14, 2015): 367–90. http://dx.doi.org/10.1002/nag.2406.

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Bazilevs, Yuri, Kenji Takizawa, and Tayfun E. Tezduyar. "Computational analysis methods for complex unsteady flow problems." Mathematical Models and Methods in Applied Sciences 29, no. 05 (May 2019): 825–38. http://dx.doi.org/10.1142/s0218202519020020.

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In this lead paper of the special issue, we provide a brief summary of the stabilized and multiscale methods in fluid dynamics. We highlight the key features of the stabilized and multiscale scale methods, and variational methods in general, that make these approaches well suited for computational analysis of complex, unsteady flows encountered in modern science and engineering applications. We mainly focus on the recent developments. We discuss application of the variational multiscale (VMS) methods to fluid dynamics problems involving computational challenges associated with high-Reynolds-number flows, wall-bounded turbulent flows, flows on moving domains including subdomains in relative motion, fluid–structure interaction (FSI), and complex-fluid flows with FSI.

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Jiang, Shan, Yue Cheng, Yao Cheng, and Yunqing Huang. "Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems." Mathematics 11, no. 24 (December 15, 2023): 4965. http://dx.doi.org/10.3390/math11244965.

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We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users’ demand. Based upon the generalized multiscale simulation on the coarse scale, the balanced truncation is applied to solve its Lyapunov equations on the reduced scale for further savings while ensuring high accuracy. A θ-implicit scheme is utilized for the fully discretization process. Finally, numerical results validate the uniform stability and robustness of our proposed method.

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Cheung, Daniel Y., Bin Duan, and Jonathan T. Butcher. "Current progress in tissue engineering of heart valves: multiscale problems, multiscale solutions." Expert Opinion on Biological Therapy 15, no. 8 (June 2015): 1155–72. http://dx.doi.org/10.1517/14712598.2015.1051527.

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Wu, Xiangyang, Haibin Shi, and Haiping Zhu. "Fault Diagnosis for Rolling Bearings Based on Multiscale Feature Fusion Deep Residual Networks." Electronics 12, no. 3 (February 3, 2023): 768. http://dx.doi.org/10.3390/electronics12030768.

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Deep learning, due to its excellent feature-adaptive capture ability, has been widely utilized in the fault diagnosis field. However, there are two common problems in deep-learning-based fault diagnosis methods: (1) many researchers attempt to deepen the layers of deep learning models for higher diagnostic accuracy, but degradation problems of deep learning models often occur; and (2) the use of multiscale features can easily be ignored, which makes the extracted data features lack diversity. To deal with these problems, a novel multiscale feature fusion deep residual network is proposed in this paper for the fault diagnosis of rolling bearings, one which contains multiple multiscale feature fusion blocks and a multiscale pooling layer. The multiple multiscale feature fusion block is designed to automatically extract the multiscale features from raw signals, and further compress them for higher dimensional feature mapping. The multiscale pooling layer is constructed to fuse the extracted multiscale feature mapping. Two famous rolling bearing datasets are adopted to evaluate the diagnostic performance of the proposed model. The comparison results show that the diagnostic performance of the proposed model is superior to not only several popular models, but also other advanced methods in the literature.

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Annunziato, M. "Analysis Of Upwind Method For Piecewise Deterministic Markov Processes." Computational Methods in Applied Mathematics 8, no. 1 (2008): 3–20. http://dx.doi.org/10.2478/cmam-2008-0001.

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AbstractA numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied.

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ILIEV, O., and I. RYBAK. "ON NUMERICAL UPSCALING FOR FLOWS IN HETEROGENEOUS POROUS MEDIA." Computational Methods in Applied Mathematics 8, no. 1 (2008): 60–76. http://dx.doi.org/10.2478/cmam-2008-0004.

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AbstractA numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied

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Dimarco, Giacomo, and Lorenzo Pareschi. "Hybrid multiscale methods for hyperbolic problems I. Hyperbolic relaxation problems." Communications in Mathematical Sciences 4, no. 1 (2006): 155–77. http://dx.doi.org/10.4310/cms.2006.v4.n1.a6.

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Madureira, Alexandre L., and Marcus Sarkis. "Hybrid Localized Spectral Decomposition for Multiscale Problems." SIAM Journal on Numerical Analysis 59, no. 2 (January 2021): 829–63. http://dx.doi.org/10.1137/20m1314896.

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26

Mielke, Alexander. "Weak-convergence methods for Hamiltonian multiscale problems." Discrete & Continuous Dynamical Systems - A 20, no. 1 (2008): 53–79. http://dx.doi.org/10.3934/dcds.2008.20.53.

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Kremer, K., and F. Müller-Plathe. "Multiscale Problems in Polymer Science: Simulation Approaches." MRS Bulletin 26, no. 3 (March 2001): 205–10. http://dx.doi.org/10.1557/mrs2001.43.

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Polymer materials range from industrial commodities, such as plastic bags, to high-tech polymers used for optical applications, and all the way to biological systems, where the most prominent example is DNA. They can be crystalline, amorphous (glasses, melts, gels, rubber), or in solution. Polymers in the glassy state are standard materials for many applications (yogurt cups, compact discs, housings for technical equipment, etc.).

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Jin, Shi. "Asymptotic-preserving schemes for multiscale physical problems." Acta Numerica 31 (May 2022): 415–89. http://dx.doi.org/10.1017/s0962492922000010.

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We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the asymptotic-preserving (AP) strategies to compute multiscale physical problems efficiently. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic these asymptotic transitions at the discrete level, and hence can be used crossing scales and, in particular, capture the macroscopic behaviour without resolving the microscopic physical scale numerically.

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Смирнов, Н. Н., В. В. Тюренкова, and В. Ф. Никитин. "Simulation Models for Solving Multiscale Combustion Problems." Успехи кибернетики / Russian Journal of Cybernetics, no. 4(8) (November 30, 2021): 30–41. http://dx.doi.org/10.51790/2712-9942-2021-2-4-3.

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Разработка алгоритмической компоновки и программ для расчета многомасштабных процессов горения является актуальной междисциплинарной темой фундаментальных исследований, которая объединяет методы информационных технологий, механики многокомпонентных сплошных сред, химии и математического моделирования. Задача разработки алгоритмической компоновки и подбора программ для расчета многомасштабных процессов горения набирает актуальность с каждым годом в связи как с интенсивным развитием вычислительных методов и моделей, так и с увеличением современных возможностей суперкомпьютерных вычислений. Практическая применимость разрабатываемых вычислительных моделей и методов охватывает проблемы энергетики, двигателестроения, взрывопожаробезопасности, а также интенсификации добычи полезных ископаемых с применением методов термохимического воздействия на пласт. Основными проблемами, возникающими в процессе моделирования, являются: а) многомасштабность, не позволяющая проводить моделирование всех задействованных процессов на единых даже масштабируемых сетках; б) жесткость и большая размерность системы дифференциальных уравнений для описания химической кинетики, решение которой может занимать 80% процессорного времени. Данная статья представляет обзор уже проведенных исследований в ФГУ ФНЦ НИИСИ РАН и анализ трудностей, с которыми столкнулись исследователи. В статье содержатся новые предложения по преодолению вычислительных трудностей и намечены пути их реализации. Возможность решения проблем в части многомасштабности видится в применении подходов многоуровневого моделирования, при котором детальное решение задачи более мелкого масштаба обрабатывается и вносится в качестве элемента модели более крупного масштаба. Для решения проблемы сокращения времени интегрирования уравнений многостадийной химической кинетики актуальным трендом является применение нейросетевых подходов и методов в рамках разрабатываемых вычислительных моделей. Этот подход в настоящее время развивается сотрудниками отдела вычислительных систем совместно с коллективом Центра оптико-нейронных технологий ФГУ ФНЦ НИИСИ РАН. The development of algorithms and software for analyzing multiscale combustion processes is a relevant field of fundamental research that combines the methods of information technologies, mechanics of multicomponent continua, combustion chemistry, and simulation. It gains relevance year to year due to the intensive development of computational methods and models, and with the increase in supercomputing performance. The applications of the proposed computational models and methods include energy, engine manufacturing, explosion and fire safety fields, as well as thermochemical mineral recovery stimulation methods. The key simulation problems are a. the problem is multiscale: all the processes involved cannot be simulated with the same grid, even a scalable one; b. the rigidity and large dimensionality of the system of differential equations that describes chemical kinetics. Its solution may take up to 80 % of the processor time. This paper is an overview of the research conducted at the Scientific Research Institute for System Analysis and an analysis of the difficulties faced by the researchers. It also proposes new ways for overcoming the computational difficulties and give some implementation considerations. To solve the multi-scale issue, multi-level modeling approaches can be used: a detailed solution to a smaller-scale problem is processed and introduced as a component of a larger-scale model. To reduce the integration time of the multi-stage chemical kinetics equations, the current approach is applying neural networks and methods to the existing computational models. This approach is currently being developed at the Department of Computing Systems in collaboration with the Center for Optical-Neural Technologies, SRISA.

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Chen, Jiong, Florian Schäfer, Jin Huang, and Mathieu Desbrun. "Multiscale cholesky preconditioning for ill-conditioned problems." ACM Transactions on Graphics 40, no. 4 (August 2021): 1–13. http://dx.doi.org/10.1145/3476576.3476637.

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Chen, Jiong, Florian Schäfer, Jin Huang, and Mathieu Desbrun. "Multiscale cholesky preconditioning for ill-conditioned problems." ACM Transactions on Graphics 40, no. 4 (August 2021): 1–13. http://dx.doi.org/10.1145/3450626.3459851.

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32

Cancès, E., and S. Labbé. "Mathematical and numerical approaches for multiscale problems." ESAIM: Proceedings 37 (September 2012): I. http://dx.doi.org/10.1051/proc/123700f.

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Wang, Yiling, Jun Hu, Kui Han, and Zaiping Nie. "Improved Algebraic Preconditioning for Multiscale Electromagnetic Problems." IEEE Antennas and Wireless Propagation Letters 16 (2017): 1447–50. http://dx.doi.org/10.1109/lawp.2016.2641927.

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Papanicolaou, Andrew, and Konstantinos Spiliopoulos. "Filtering the Maximum Likelihood for Multiscale Problems." Multiscale Modeling & Simulation 12, no. 3 (January 2014): 1193–229. http://dx.doi.org/10.1137/140952648.

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Limon, Alfonso, and Hedley Morris. "Multiscale cell-based coarsening for discontinuous problems." Mathematics and Computers in Simulation 79, no. 6 (February 2009): 1915–25. http://dx.doi.org/10.1016/j.matcom.2007.06.011.

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Bellomo, N., Y. Tao, and M. Winkler. "Cross-diffusion models: Analytic and multiscale problems." Mathematical Models and Methods in Applied Sciences 28, no. 11 (October 2018): 2097–102. http://dx.doi.org/10.1142/s0218202518020025.

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A presentation of a special issue on the derivation of cross-diffusion models and on the related analytical problems is proposed in this note. A brief introduction to motivations and recently published literature is presented in the first part. Subsequently, a concise description of the contents of the papers published in the issue follows. Finally, some ideas on possible research perspectives are proposed.

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Fang, Ying-guang, and Bo Li. "Multiscale problems and analysis of soil mechanics." Mechanics of Materials 103 (December 2016): 55–67. http://dx.doi.org/10.1016/j.mechmat.2016.09.003.

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Ozer, Gungor, Antoni Luque, and Tamar Schlick. "The chromatin fiber: multiscale problems and approaches." Current Opinion in Structural Biology 31 (April 2015): 124–39. http://dx.doi.org/10.1016/j.sbi.2015.04.002.

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Elfverson, Daniel, Mats G. Larson, and Axel Målqvist. "Multiscale methods for problems with complex geometry." Computer Methods in Applied Mechanics and Engineering 321 (July 2017): 103–23. http://dx.doi.org/10.1016/j.cma.2017.03.023.

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Ladeveze, P., A. Nouy, and O. Loiseau. "A multiscale computational approach for contact problems." Computer Methods in Applied Mechanics and Engineering 191, no. 43 (September 2002): 4869–91. http://dx.doi.org/10.1016/s0045-7825(02)00406-1.

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Hu, Nan, and Jacob Fish. "Enhanced ant colony optimization for multiscale problems." Computational Mechanics 57, no. 3 (January 2, 2016): 447–63. http://dx.doi.org/10.1007/s00466-015-1245-z.

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Abdulle, Assyr, Giacomo Garegnani, and Andrea Zanoni. "Ensemble Kalman Filter for Multiscale Inverse Problems." Multiscale Modeling & Simulation 18, no. 4 (January 2020): 1565–94. http://dx.doi.org/10.1137/20m1348431.

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Zabaras, Nicholas, and Dongbin Xiu. "Stochastic modeling of multiscale and multiphysics problems." Computer Methods in Applied Mechanics and Engineering 197, no. 43-44 (August 2008): 3419. http://dx.doi.org/10.1016/j.cma.2008.04.021.

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Sun, WaiChing, Zhijun Cai, and Jinhyun Choo. "Mixed Arlequin method for multiscale poromechanics problems." International Journal for Numerical Methods in Engineering 111, no. 7 (February 16, 2017): 624–59. http://dx.doi.org/10.1002/nme.5476.

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Sharma, Somya, Marten Thompson, Debra Laefer, Michael Lawler, Kevin McIlhany, Olivier Pauluis, Dallas R. Trinkle, and Snigdhansu Chatterjee. "Machine Learning Methods for Multiscale Physics and Urban Engineering Problems." Entropy 24, no. 8 (August 16, 2022): 1134. http://dx.doi.org/10.3390/e24081134.

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We present an overview of four challenging research areas in multiscale physics and engineering as well as four data science topics that may be developed for addressing these challenges. We focus on multiscale spatiotemporal problems in light of the importance of understanding the accompanying scientific processes and engineering ideas, where “multiscale” refers to concurrent, non-trivial and coupled models over scales separated by orders of magnitude in either space, time, energy, momenta, or any other relevant parameter. Specifically, we consider problems where the data may be obtained at various resolutions; analyzing such data and constructing coupled models led to open research questions in various applications of data science. Numeric studies are reported for one of the data science techniques discussed here for illustration, namely, on approximate Bayesian computations.

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Fu, Ping, Hui Liu, Xihua Chu, and Yuanjie Xu. "A Multiscale Computational Formulation for Gradient Elasticity Problems of Heterogeneous Structures." International Journal of Computational Methods 13, no. 05 (August 31, 2016): 1650030. http://dx.doi.org/10.1142/s0219876216500304.

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In this paper, a multiscale computational formulation is developed for modeling two- and three-dimensional gradient elasticity behaviors of heterogeneous structures. To capture the microscopic properties at the macroscopic level effectively, a numerical multiscale interpolation function of coarse element is constructed by employing the oversampling element technique based on the staggered gradient elasticity scheme. By virtue of these functions, the equivalent quantities of the coarse element could be obtained easily, resulting in that the material microscopic characteristics are reflected to the macroscopic scale. Consequently, the displacement field of the original boundary value problem could be calculated at the macroscopic level, and the corresponding microscopic gradient-enriched solutions could also be evaluated by adopting the downscaling computation on the sub-grids of each coarse element domain, which will reduce the computational cost significantly. Furthermore, several representative numerical experiments are performed to demonstrate the validity and efficiency of the proposed multiscale formulation.

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Zhang, Ping, and Xiaohua Zhang. "Numerical Modeling of Stokes Flow in a Circular Cavity by Variational Multiscale Element Free Galerkin Method." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/451546.

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The variational multiscale element free Galerkin method is extended to simulate the Stokes flow problems in a circular cavity as an irregular geometry. The method is combined with Hughes’s variational multiscale formulation and element free Galerkin method; thus it inherits the advantages of variational multiscale and meshless methods. Meanwhile, a simple technique is adopted to impose the essential boundary conditions which makes it easy to solve problems with complex area. Finally, two examples are solved and good results are obtained as compared with solutions of analytical and numerical methods, which demonstrates that the proposed method is an attractive approach for solving incompressible fluid flow problems in terms of accuracy and stability, even for complex irregular boundaries.

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SUZUKI, Yoshiro, Akira TODOROKI, and Yoshihiro MIZUTANI. "Multiscale seamless-domain method for solving nonlinear heat conduction problems without iterative multiscale calculations." Mechanical Engineering Journal 3, no. 4 (2016): 15–00491. http://dx.doi.org/10.1299/mej.15-00491.

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Deng, Weibing, and Haijun Wu. "A Combined Finite Element and Multiscale Finite Element Method for the Multiscale Elliptic Problems." Multiscale Modeling & Simulation 12, no. 4 (January 2014): 1424–57. http://dx.doi.org/10.1137/120898279.

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Chen, Yanping, and Yuelong Tang. "Numerical Methods for Constrained Elliptic Optimal Control Problems with Rapidly Oscillating Coefficients." East Asian Journal on Applied Mathematics 1, no. 3 (August 2011): 235–47. http://dx.doi.org/10.4208/eajam.071010.250411a.

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AbstractIn this paper we use two numerical methods to solve constrained optimal control problems governed by elliptic equations with rapidly oscillating coefficients: one is finite element method and the other is multiscale finite element method. We derive the convergence analysis for those two methods. Analytical results show that finite element method can not work when the parameter ε is small enough, while multiscale finite element method is useful for any parameter ε.

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Journal articles on the topic 'Multiscale problems'

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