【摘要】：有限體積元方法又稱為控制體積法,盒式方法,廣義差分法,是在有限差分方法和有限元方法的基礎(chǔ)上發(fā)展起來的求解偏微分方程的重要數(shù)值方法.有限體積元方法既保持了有限差分方法便于計算的優(yōu)點,又保持了有限元方法的計算精度,同時又具有局部守恒性質(zhì),這使得該方法在流體力學(xué)和熱傳導(dǎo)方程等諸多領(lǐng)域上獲得廣泛的關(guān)注.本文著重研究了二階橢圓問題的自適應(yīng)有限體積元方法的多重網(wǎng)格算法,即在網(wǎng)格加密過程中不再進行一致加密,只在后驗誤差較大的區(qū)域利用最新頂點二分法進行局部加密,從而使得問題的自由度大幅降低,減少問題的計算量,提高計算的精度.在處理問題時,我們可以將有限體積元方法看作是有限元方法的一個擾動,然后依據(jù)自適應(yīng)有限元方法多重網(wǎng)格算法的收斂性結(jié)果,建立自適應(yīng)有限體方法的相關(guān)收斂性理論.文章首先簡單介紹了自適應(yīng)有限體積元方法的國內(nèi)外研究現(xiàn),和本文選題的研究意,其次敘述了最新頂點二分法的步驟,有限體積元方法的構(gòu)造思想和多重網(wǎng)格的相關(guān)理論.最后文章證明了問題的收斂性分析結(jié)果,并給出數(shù)值算例來驗證理論的結(jié)果.
[Abstract]:The finite volume element method, also known as the control volume method, the box method and the generalized difference method, is an important numerical method to solve the partial differential equation based on the finite difference method and the finite element method. The finite volume element method not only maintains the advantages of the finite difference method, but also keeps the calculation precision of the finite element method. In this paper, the multigrid algorithm of adaptive finite volume element method for two order elliptic problems is studied in this paper, that is, no uniform encryption is carried out in the process of grid encryption, only the posteriori error is larger. The region uses the latest vertex dichotomy for local encryption so that the degree of freedom of the problem is greatly reduced, the computational complexity of the problem is reduced, and the accuracy of the calculation is improved. The convergence theory of the adaptive finite body method is established. First, the paper briefly introduces the research at home and abroad of the adaptive finite volume element method and the research meaning of this topic. Secondly, it describes the steps of the latest vertex dichotomy, the construction of the finite volume element method and the related theory of the multigrid. Finally, the theory of the finite volume element method and the related theory of the multigrid are also described. The convergence of the problem is proved and a numerical example is given to verify the theoretical results.
【學(xué)位授予單位】：煙臺大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.82

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本文編號：2152530