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研究生: 高偉庭
Kao, Wei-Ting
論文名稱: 一個解非線性斯托克斯方程的弱伽遼金有限元素法應用於冰層動力學
A weak Galerkin finite element method for solving the nonlinear Stokes equation with application to ice-sheet dynamics
指導教授: 朱家杰
Chu, Chia-Chieh Jay
口試委員: 蔡志強
Tsai, Je-Chiang
薛名成
Shiue, Ming-Cheng
學位類別: 碩士
Master
系所名稱: 理學院 - 數學系
Department of Mathematics
論文出版年: 2022
畢業學年度: 110
語文別: 英文
論文頁數: 26
中文關鍵詞: 有限元素法非線性斯托克斯方程弱伽遼金方法冰層動力學
外文關鍵詞: finite element methods, nonlinear Stokes equation, weak Galerkin methods, ice-sheet dynamics
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  • 近年來,由於全球氣溫上升,兩極的冰川開始被大家重視。與其相關的
    科學領域是冰層動力學,一門探討冰層受到外力後如何運動的自然科學。
    在數學領域上,冰層模型多用非線性斯托克斯方程描述。我們使用弱伽遼
    金有限元素法求解此偏微分方程,並且利用相同方法下解線性托克斯方程
    的結果及固定點疊代法,給出有效的估計結果。


    Recently, due to global temperature raise, the polar glaciers are gradually emphasized.
    The related science field is glacier dynamics, describe how glaciers move
    under the environmental force. The glacier model is usually modelled by nonlinear
    Stokes equation in mathematics. We use weak Galerkin finite element method
    to solve this PDE and give an effective approximation by the fixed point iteration
    method and the result of solving the linear Stokes equation in the weak Galerkin
    finite element method.

    Acknowledgements 摘要i Abstract ii 1 Introduction 1 1.1 Motivation 1 1.2 The model of glaciers 2 1.3 The Stokes equation 3 2 Method and schemes 7 2.1 Finite element space 7 2.2 Implementation of the method 10 2.3 Error estimate 11 3 Numerical Experiences 15 3.1 Linear case Result: variable coefficient 15 3.2 Nonlinear case Result: Main case 18 4 Conclusion and Future works 23 References 25

    [1] R. DeConto and D. Pollard, “Contribution of antarctica to past and future sea-level rise,”
    Nature, vol. 531, pp. 591–597, 03 2016.
    [2] J. Caman͂o, G. N. Gatica, R. Oyarzúa, and G. Tierra, “An augmented mixed finite element
    method for the navier–stokes equations with variable viscosity,” SIAM Journal on
    Numerical Analysis, vol. 54, no. 2, pp. 1069–1092, 2016.
    [3] L. Mu and X. Ye, “A simple finite element method for the Stokes equations,” Adv. Comput.
    Math., vol. 43, no. 6, pp. 1305–1324, 2017.
    [4] G. Cheng, P. Lötstedt, and L. von Sydow, “Accurate and stable time stepping in ice sheet
    modeling,” Journal of Computational Physics, vol. 329, pp. 29–47, jan 2017.
    [5] K. Hutter, The Application of the Shallow-Ice Approximation, pp. 256–332. Dordrecht:
    Springer Netherlands, 1983.
    [6] E. Bueler, Numerical Modelling of Ice Sheets, Streams, and Shelves, pp. 185–217. 01
    2021.
    [7] E. Bueler and J. Brown, “Shallow shelf approximation as a “sliding law"in a thermomechanically
    coupled ice sheet model,” Journal of Geophysical Research: Earth Surface,
    vol. 114, no. F3, 2009.
    [8] Q. Chen, M. Gunzburger, and M. Perego, “Well-posedness results for a nonlinear stokes
    problem arising in glaciology,” SIAM Journal on Mathematical Analysis, vol. 45, no. 5,
    pp. 2710–2733, 2013.
    [9] D. Sanchez, L. Hume, R. Chatelin, and P. Poncet, “Analysis of 3D non-linear Stokes problem
    coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications
    to digital rock physics and mucociliary clearance,” ESAIM: Mathematical Modelling
    and Numerical Analysis, vol. 53, no. 4, pp. 1083–1124, 2019.
    [10] J. Djoko, J. Koko, and R. Kucera, “Power law stokes equations with threshold slip boundary
    conditions: Numerical methods and implementation,” Mathematical Methods in the
    Applied Sciences, vol. 42, 03 2019.
    [11] I. S. Monnesland, E. Lee, M. Gunzburger, and R. Yoon, “A least-squares finite element
    method for a nonlinear Stokes problem in glaciology,” Comput. Math. Appl., vol. 71,
    no. 11, pp. 2421–2431, 2016.
    [12] The Construction of a Finite Element Space, pp. 69–92. New York, NY: Springer New
    York, 2008.
    [13] P. A. Raviart and J. M. Thomas, “A mixed finite element method for 2-nd order elliptic
    problems,” in Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes,
    eds.), (Berlin, Heidelberg), pp. 292–315, Springer Berlin Heidelberg, 1977.
    [14] L. Mu, J. Wang, Y. Wang, and X. Ye, “A weak Galerkin mixed finite element method
    for biharmonic equations,” in Numerical solution of partial differential equations: theory,
    algorithms, and their applications, vol. 45 of Springer Proc. Math. Stat., pp. 247–277,
    Springer, New York, 2013.
    [15] J. Wang and X. Ye, “A weak galerkin finite element method for the stokes equations,”
    2013.
    [16] J. Wang and X. Ye, “A weak Galerkin mixed finite element method for second order elliptic
    problems,” Math. Comp., vol. 83, no. 289, pp. 2101–2126, 2014.
    [17] L. Mu, “Pressure robust weak Galerkin finite element methods for Stokes problems,”
    SIAM J. Sci. Comput., vol. 42, no. 3, pp. B608–B629, 2020.

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