An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems
Eric T. Chung, Yalchin R. Efendiev, Wing Tat Leung
In this paper, we develop an adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for a class of high-contrast flow problems and derive a priori and a posteriori error estimates for the method. Based on the a posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM Rev., 43 (2001), pp. 129-159]). The proposed error indicators are L-based and can be inexpensively computed, which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.
Multiscale Modeling & Simulation
https://doi.org/10.1137/140986189
https://repository.kaust.edu.sa/bitstream/10754/631503/1/140986189.pdf
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Galerkin method Earth & Environmental Sciences 100%
Galerkin methods Chemical Compounds 88%
Discontinuous Galerkin Method Mathematics 86%
Basis Functions Mathematics 55%
Error Indicator Mathematics 52%
indicator Earth & Environmental Sciences 31%
A Posteriori Error Estimators Mathematics 23%
Approximation Space Mathematics 22%
Chung, E. T., Efendiev, Y. R., & Leung, W. T. (2018). An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems. Multiscale Modeling & Simulation, 16(3), 1227-1257. https://doi.org/10.1137/140986189
Chung, Eric T. ; Efendiev, Yalchin R. ; Leung, Wing Tat. / An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems. In: Multiscale Modeling & Simulation. 2018 ; Vol. 16, No. 3. pp. 1227-1257.
@article{7a9f34a4a9a84d11851f71d534d5761f,
title = "An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems",
abstract = "In this paper, we develop an adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for a class of high-contrast flow problems and derive a priori and a posteriori error estimates for the method. Based on the a posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM Rev., 43 (2001), pp. 129-159]). The proposed error indicators are L-based and can be inexpensively computed, which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.",
author = "Chung, {Eric T.} and Efendiev, {Yalchin R.} and Leung, {Wing Tat}",
note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The first author was partially supported by the Hong Kong RGC General Research Fund (project 400813) and CUHK Direct Grant for Research 2013/14.",
doi = "10.1137/140986189",
journal = "Multiscale Modeling & Simulation",
publisher = "Society for Industrial and Applied Mathematics Publications",
Chung, ET, Efendiev, YR & Leung, WT 2018, 'An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems', Multiscale Modeling & Simulation, vol. 16, no. 3, pp. 1227-1257. https://doi.org/10.1137/140986189
An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems. / Chung, Eric T.; Efendiev, Yalchin R.; Leung, Wing Tat.
In: Multiscale Modeling & Simulation, Vol. 16, No. 3, 07.08.2018, p. 1227-1257.
T1 - An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems
AU - Chung, Eric T.
AU - Efendiev, Yalchin R.
AU - Leung, Wing Tat
N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The first author was partially supported by the Hong Kong RGC General Research Fund (project 400813) and CUHK Direct Grant for Research 2013/14.
N2 - In this paper, we develop an adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for a class of high-contrast flow problems and derive a priori and a posteriori error estimates for the method. Based on the a posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM Rev., 43 (2001), pp. 129-159]). The proposed error indicators are L-based and can be inexpensively computed, which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.
AB - In this paper, we develop an adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for a class of high-contrast flow problems and derive a priori and a posteriori error estimates for the method. Based on the a posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM Rev., 43 (2001), pp. 129-159]). The proposed error indicators are L-based and can be inexpensively computed, which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.
UR - https://epubs.siam.org/doi/10.1137/140986189
U2 - 10.1137/140986189
DO - 10.1137/140986189
JO - Multiscale Modeling & Simulation
JF - Multiscale Modeling & Simulation
Chung ET, Efendiev YR, Leung WT. An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems. Multiscale Modeling & Simulation. 2018 Aug 7;16(3):1227-1257. https://doi.org/10.1137/140986189