本文選題：雜交間斷伽略金方法 + 復雜界面��； 參考：《湖南師范大學》2016年博士論文

【摘要】：本學位論文提出并分析了一種求解界面問題的一致網(wǎng)格方法。首先,本文以泊松界面問題為例給出了該算法的基本思想。通過在界面附近構(gòu)造一個新的分片多項式函數(shù),得到一個界面與網(wǎng)格的邊界一致的擴展界面問題。然后,采用雜交間斷伽略金有限元方法(HDG)來求解該擴展問題,通過合理地選擇數(shù)值通量,解在單元邊界上的跳躍被自然地引入到了數(shù)值格式當中。與現(xiàn)有的方法相比,該方法構(gòu)造的分片多項式函數(shù)是通過一個巧妙的二次Hermite多項式插值,外加一個標準的拉格朗日多項式插值的后處理得到。上述顯式構(gòu)造的多項式能夠準確地捕捉到界面上的跳躍信息,而且存在唯一,并以3階精度逼近原問題精確解的間斷部分,更重要的是它與界面的形狀及位置無關(guān)。另外,它使得擴展界面問題的解具有較高的正則性,從而保證了我們得以用HDG方法高精度求解。最后證明了該方法在L2范數(shù)意義下,其精確解及梯度均具有二階收斂精度。文中涉及各種復雜界面橢圓問題的數(shù)值例子也驗證了該算法的穩(wěn)定性及收斂性�；谇蠼獠此山缑鎲栴}的成功經(jīng)驗,本文隨后研究了拋物界面問題,并重點考慮了移動界面情形。由于界面的位置和形狀隨著時間不斷變化,因此需要在每一個時刻構(gòu)造逼近解的間斷部分的高精度分片多項式,在此基礎(chǔ)上,它將原問題轉(zhuǎn)化成界面與網(wǎng)格的邊界重合的擴展界面問題。之后,利用HDG方法對擴展界面問題進行空間離散。通過合理設計數(shù)值通量,解在單元邊界上的跳躍被自然地引入到了數(shù)值格式當中,從而保證了離散格式的二階收斂精度。在時間方向,本文采用經(jīng)典的向后歐拉格式進行離散,以保證全離散格式的數(shù)值穩(wěn)定性。值得指出的是,每一時刻構(gòu)造解的間斷部分的分片多項式逼近的方法是不變的,只是界面的位置以及界面上的跳躍條件發(fā)生了變化,而這種變化只會對每一步要求解的線性方程組的右端產(chǎn)生影響,并不會改變線性方程組的系數(shù)矩陣。因此,在計算時只需要在第一個時間步完成對線性方程組系數(shù)矩陣的計算和組裝,在后面所有時間步,只需要反復使用已經(jīng)組裝好的系數(shù)矩陣,從而大幅度提高了算法的計算效率。大量數(shù)值實驗表明,在笛卡爾網(wǎng)格下,隨著界面的移動,該方法不僅穩(wěn)定,而且能夠保證解及其梯度在L2范數(shù)意義下具有二階收斂精度。在對泊松界面問題的研究中,出于理論分析的考慮,只討論了帶有形如[[%絬·n]]仿射跳躍條件的界面問題。為了處理更一般的帶間斷系數(shù)的界面問題,本文引進一種迭代技巧。通過該迭代技巧,一般的帶有間斷系數(shù)的界面問題的解,可以被一系列帶有簡單仿射跳躍條件的界面問題的解逼近,并且只要選取適當?shù)氖諗恳蜃蛹纯杀ＷC這種迭代法的收斂性。因此我們只需對這一系列逼近問題采用本文所提出的數(shù)值方法就可以實現(xiàn)對一般界面問題的高精度求解。與移動界面問題的求解類似,每一個逼近的界面問題,需要根據(jù)其解所滿足的跳躍條件構(gòu)造相應的解的間斷部分的分片多項式逼近。為了驗證算法的有效性,我們考察了圓環(huán)區(qū)域上帶有一階吸收邊界條件的Helmholtz界面問題。數(shù)值實驗表明,在擬一致網(wǎng)格下,該數(shù)值方法不僅穩(wěn)定,而且解及其梯度在L2范數(shù)的意義下皆具有二階收斂精度。
[Abstract]:In this thesis, a uniform grid method for solving interface problems is proposed and analyzed. First, the basic idea of this algorithm is given as an example of the Poisson interface problem. By constructing a new piecewise polynomial function near the interface, an extended interface problem with the boundary of the interface to the grid is obtained. The discontinuous Galerkin finite element method (HDG) is used to solve the extension problem. By selecting the numerical flux reasonably, the jump is naturally introduced into the numerical scheme in the element boundary. Compared with the existing method, the piecewise polynomial function constructed by this method is interpolated by a clever two times Hermite polynomial, plus a standard. The postprocessing of the quasi Lagrange polynomial interpolation is obtained. The polynomial of the above explicit construction can accurately capture the jumping information on the interface, and there is a unique, and the 3 order accuracy approximates the discontinuous part of the exact solution of the original problem, and more importantly, it is independent of the shape and position of the interface. In addition, it makes the solution of the extended interface problem. It has high regularity, which ensures that we can solve the high precision by HDG method. Finally, it is proved that the method has two order convergence precision in the sense of L2 norm. The numerical examples of various complex interface elliptic problems in this paper also verify the stability and convergence of the method. After the successful experience of the surface problem, this paper studies the problem of the parabolic interface and focuses on the mobile interface. Because the position and shape of the interface vary with time, the high precision piecewise polynomial of the discontinuous part of the solution is constructed at every moment. On this basis, it transforms the original problem into the interface and the net. The extended interface problem of the boundary coincidence of the lattice is solved. After that, the HDG method is used to discrete the extended interface problem. By reasonably designing the numerical flux, the jump is naturally introduced into the numerical scheme, thus ensuring the two order convergence accuracy of the discrete scheme. In the time direction, the classical backward direction is used in this paper. The Euler scheme is discrete to ensure the numerical stability of the full discrete scheme. It is worth noting that the piecewise polynomial approximation of the discontinuous part of the structural solution at every moment is invariable, only the position of the interface and the jumping conditions on the interface change, and this change will only be a linear equation set for each step. The right end has an influence and does not change the coefficient matrix of the linear equations. Therefore, the calculation and assembly of the linear equation group coefficient matrix is only needed in the first time step. In all the time steps, only the coefficient matrix which has been assembled is used repeatedly, which greatly improves the computational efficiency of the algorithm. Numerical experiments show that in Cartesian grid, with the movement of the interface, the method is not only stable, but also can guarantee the two order convergence precision of the solution and its gradient in the sense of L2 norm. In the study of the Poisson interface problem, only the interface problem with the shape like [%] n]] affine jumping condition is discussed for the consideration of the theoretical analysis. In order to deal with more general interface problems with discontinuous coefficients, an iterative technique is introduced in this paper. Through this iterative technique, the general solution of interface problems with discontinuous coefficients can be approximated by a series of solutions with simple affine jumping conditions, and the iterative method can be guaranteed by selecting the appropriate convergence factor. Therefore, we only need to use the numerical method proposed in this series of approximation problems to achieve a high precision solution to the general interface problem. It is similar to the solution of the mobile interface problem. The interface problem of each approximation needs to be divided into the discontinuous parts of the corresponding solution according to the jump conditions satisfied by the solution. In order to verify the validity of the algorithm, in order to verify the effectiveness of the algorithm, we examine the Helmholtz interface problem with the first order absorption boundary conditions in the ring region. The numerical experiment shows that the numerical method is not only stable but also has two order convergence precision in the sense of L2 norm.
【學位授予單位】：湖南師范大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O241.82

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