本文選題：周期初始條件 + 一維雙曲微分方程��； 參考：《湖南科技大學(xué)》2017年碩士論文

【摘要】：間斷有限元方法是使用完全不連續(xù)的分片多項(xiàng)式空間作為解空間和檢驗(yàn)函數(shù)空間的一類(lèi)有限元方法,間斷有限元法解偏微分方程的超收斂性質(zhì)也是最近幾年來(lái)本研究領(lǐng)域?qū)W者們非常感興趣的研究主題。本文研究了求解一類(lèi)一維具有周期初始條件的微分方程間斷有限元計(jì)算方法及其收斂性質(zhì)。本文主要研究了具有周期初始條件的一階雙曲微分方程和拋物方程定解問(wèn)題。對(duì)于一般的一階雙曲方程,首先將其轉(zhuǎn)化等價(jià)具有周期邊界的混合邊界問(wèn)題,研究了選取迎風(fēng)數(shù)值流量時(shí)對(duì)應(yīng)的有限元方法,構(gòu)造校正函數(shù)得到超逼近有限元的插值函數(shù),依次證明了一次間斷有限元和任意間斷有限元的逐點(diǎn)誤差以及區(qū)間平均值誤差估計(jì);其次推導(dǎo)了一次有限元的時(shí)間向前全離散計(jì)算格式和向后全離散計(jì)算格式、二次有限元的4階Runge-Kutta全離散計(jì)算格式;最初給出了兩個(gè)數(shù)值例子驗(yàn)證了計(jì)算方法的有效性。對(duì)于一般的拋物方程定解問(wèn)題,簡(jiǎn)單介紹了局部間斷元方法,并推導(dǎo)了一次元的時(shí)間向前全離散計(jì)算格式和向后全離散計(jì)算格式;二次有限元的4階Runge-Kutta全離散計(jì)算格式。
[Abstract]:Discontinuous finite element method is a kind of finite element method which uses completely discontinuous piecewise polynomial space as solution space and test function space. The superconvergence property of discontinuous finite element method for solving partial differential equations is also a subject of great interest to scholars in this field in recent years. In this paper, the discontinuous finite element method for solving a class of one-dimensional differential equations with periodic initial conditions and its convergence properties are studied. In this paper, we study the solutions of first order hyperbolic differential equations and parabolic equations with periodic initial conditions. For a general first order hyperbolic equation, the mixed boundary problem with periodic boundary is transformed into a mixed boundary problem. The finite element method corresponding to the selection of upwind numerical flux is studied, and a correction function is constructed to obtain the interpolation function of the superapproximate finite element. The point-by-point error and interval mean error estimation of one-order discontinuous finite element and arbitrary discontinuous finite element are proved in turn. The fourth order Runge-Kutta full discrete scheme of quadratic finite element method is presented, and two numerical examples are given to verify the validity of the method. For general parabolic equations, the local discontinuous element method is briefly introduced, and the time forward and backward full discrete schemes of the first order element and the fourth order Runge-Kutta full discrete scheme of quadratic finite element are derived.
【學(xué)位授予單位】：湖南科技大學(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)：1956715