In this thesis, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which is the norm of a local residual operator. Based on this indicator, we have an offiine adaptive method to increase the number of basis functions on the coarse grid edges with large local error estimates. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution. We show theoretically and numerically the convergence of the two methods. The online method is in general better than the offiine method, and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the online method can be independent of the contrast of the medium. / 在本文中，我們為混合廣義多尺度有限元法( mixed generalized multiscale finite element method) 在非均勻介質解決流動問題的應用上提出了兩種自適應的擴張基的方法。我們以公式局部餘差的範數制定了一個後驗誤差的指標。基於這個指標，我們得出一個離線的自適應方法。這個方法遂步把在指標數值較大的組網內邊鄰上定義的函數加入基。我們亦制定了一個在線的自適應方法。這個方法反復地將一些基於上一步驟所得的解去計算的函數加入基里，以擴大函數空間。我們以理論和實驗去證明兩個方法的收斂性質。一般而育，在線方法的收斂比離線方法快，而兩者的收斂速度都比均勻地擴充基所得的收斂速度快。從理論所得，在線方法需要一定數量的初始基函數才能達至最理想的收斂速度。實驗結果證實了這一點，並進一步得出，假如初始基函數的數目足夠，在線方法的收斂性質會不受介質的對比度影響。 / Chan, Ho Yuen. / Thesis M.Phil. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 44-46). / Abstracts also in Chinese. / Title from PDF title page (viewed on 11, October, 2016). / Detailed summary in vernacular field only.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_1291502 |
| Date | January 2015 |
| Contributors | Chan, Ho Yuen (author.), Chung, Tsz Shun Eric (thesis advisor.), Chinese University of Hong Kong Graduate School. Division of Mathematics. (degree granting institution.) |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography, text |
| Format | electronic resource, electronic resource, remote, 1 online resource (iv, 46 leaves) : illustrations (some color), computer, online resource |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons "Attribution-NonCommercial-NoDerivatives 4.0 International" License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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