【摘要】：有限體積元方法(FVEM)是求解偏微分方程的一類(lèi)重要的數(shù)值方法.它可以被視為廣義的有限差分方法,也可以看作一種廣義的Petrov-Galerkin方法.該方法的關(guān)鍵之處在于控制體單元以及測(cè)試函數(shù)空間、檢驗(yàn)函數(shù)空間的選取.間斷有限體積元方法(DFVEM)可以被視為有限體積元方法與間斷Galerkin方法的一種耦合,方法不需要逼近函數(shù)在單元邊界處連續(xù).本文考慮求解帶有振蕩系數(shù)的多尺度橢圓問(wèn)題.由于系數(shù)的振蕩性質(zhì),間斷有限體積元方法無(wú)法準(zhǔn)確地求解多尺度問(wèn)題.超樣本多尺度基函數(shù)可以有效地抓住解的多尺度信息.本文考慮基于超樣本多尺度基函數(shù)的間斷有限體積元方法求解多尺度問(wèn)題,稱(chēng)之為多尺度間斷有限體積元方法(MsDFVEM).本文給出的多尺度間斷有限體積元方法,可以被視為多尺度間斷Petrov-Galerkin方法(MsDPGM)的擾動(dòng).結(jié)合已有MsDPGM的結(jié)論,以及對(duì)擾動(dòng)項(xiàng)的估計(jì),在周期系數(shù)情形下,我們給出了 MsDFVEM嚴(yán)格的理論分析.最后,我們通過(guò)數(shù)值實(shí)驗(yàn)驗(yàn)證了方法的準(zhǔn)確性和有效性.
[Abstract]:Finite volume element method (FVEM) is an important numerical method for solving partial differential equations. It can be regarded as a generalized finite difference method or a generalized Petrov-Galerkin method. The key points of this method are the control unit, the test function space and the selection of the function space. The discontinuous finite volume element method (DFVEM) can be regarded as a coupling between the finite volume element method and the discontinuous Galerkin method. In this paper, we consider solving multi-scale elliptic problems with oscillatory coefficients. Because of the oscillation property of the coefficients, the discontinuous finite volume element method can not solve the multi-scale problem accurately. The hypersample multiscale basis function can effectively capture the multi-scale information of the solution. In this paper, the discontinuous finite volume element method based on the supersample multiscale basis function is considered to solve the multi-scale problem, which is called the multi-scale discontinuous finite volume element method (MsDFVEM). The multiscale discontinuous finite volume element method presented in this paper can be regarded as the perturbation of the (MsDPGM) method of the multiscale discontinuous Petrov-Galerkin method. Combined with the existing conclusions of MsDPGM and the estimation of the perturbation term, we give a rigorous theoretical analysis of MsDFVEM in the case of periodic coefficients. Finally, the accuracy and validity of the method are verified by numerical experiments.
【學(xué)位授予單位】：南京大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.82
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本文編號(hào)：2239644