一類非局部廣義彈性模型混合形式的有限元數(shù)值方法
發(fā)布時間：2018-12-17 03:22
【摘要】：本文考慮如下一類非局部廣義彈性模型:其中,Ω =(0,1),d~+0和d~-0分別表示左右擴(kuò)散系數(shù),表示源匯項,0D_y~β和yD_1~β分別表示β階左、右Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)算子.在此模型中,同時出現(xiàn)了積分運算和求導(dǎo)運算,于是,引入一個中間變量來求解此類積分微分方程的想法是自然的.通過令被積函數(shù)中分?jǐn)?shù)階導(dǎo)數(shù)的部分作為中間變量,我們將原問題分解為一個1-α階積分方程和一個β階微分方程,由此可以定義其混合形式的變分格式.因為得到的兩個等價方程不需要滿足一定的耦合關(guān)系,可獨立求解,我們只需證明雙線性形式在空間H-(1-α)/2(Ω)×H0β/2(Ω)中具有強(qiáng)制性和連續(xù)性,根據(jù)Lax-Milgram引理即可得到混合問題的變分解的適定性.在分?jǐn)?shù)階積分方程解的適定性的討論中,我們也得到了一種關(guān)于一類第一型Fredholm積分方程在空間H-(1-α)/2(Ω)中可解性的判定準(zhǔn)則.基于混合形式的變分原理,進(jìn)一步定義了混合形式的有限元離散格式,并證明了此格式數(shù)值解的存在唯一性.針對這一數(shù)值模擬,我們利用插值算子和L2投影算子的誤差估計性質(zhì)分別給出了關(guān)于中間變量和最終變量的能量模估計.數(shù)值試驗的結(jié)果驗證了此格式的準(zhǔn)確性.由于分?jǐn)?shù)階算子具有非局部性質(zhì),在由此得到的離散格式中,線性方程組的系數(shù)矩陣多為稠密矩陣.對于一個N階問題而言,矩陣的存儲量為O(N2),直接求解(如Gauss消元法)的計算量為O(N~3),伴隨著N的增大,問題的復(fù)雜度將使得計算時間過長而喪失了算法的高效性.于是,我們要為此類問題的求解尋找一種實現(xiàn)加速的計算方法.當(dāng)我們選擇分片常數(shù)多項式函數(shù)和分片線性多項式函數(shù)分別近似中間變量和最終變量時,經(jīng)過計算發(fā)現(xiàn),與離散格式相對應(yīng)的系數(shù)矩陣具有或部分具有Toeplitz結(jié)構(gòu).我們知道,Toeplitz矩陣的存儲量可降低為O(N),且Toeplitz矩陣-向量積的計算量為O(N log N),因此,我們可以在共軛梯度法的基礎(chǔ)上設(shè)計一種求解此類線性方程組的快速算法,使得矩陣的存儲量為O(N),每步迭代的計算量為O(N log N).對一些條件數(shù)不好的矩陣而言,加入合適的預(yù)處理子可以進(jìn)一步減少迭代次數(shù)從而提高計算效率.數(shù)值試驗的結(jié)果驗證了此快速算法的有效性.
[Abstract]:In this paper, we consider a class of nonlocal generalized elastic models, where 惟 = (0 ~ 1), d ~ 0 and d ~ 0 denote the left and right diffusivity coefficient, denote the source term, and denote the left and right Riemann-Liouville fractional derivative operators of 尾 order respectively. In this model, there are integral operations and derivation operations at the same time, so it is natural to introduce an intermediate variable to solve this kind of integrodifferential equation. By taking the fractional derivative part of the integrable function as the intermediate variable, we decompose the original problem into an integral equation of order 1- 偽 and a differential equation of order 尾, and then define its mixed form variational scheme. Because the two equivalent equations do not need to satisfy a certain coupling relation and can be solved independently, we only need to prove that the bilinear form is mandatory and continuous in the space H- (1- 偽) / 2 (惟) 脳 H0 尾 / 2 (惟). According to the Lax-Milgram 's Lemma, we can obtain the proper definiteness of the variational decomposition of the mixed problem. In the discussion of the fitness of solutions for fractional integral equations, we also obtain a criterion for the solvability of a class of first type Fredholm integral equations in space H- (1- 偽) / 2 (惟). Based on the variational principle of mixed form, the finite element discrete scheme of mixed form is further defined, and the existence and uniqueness of the numerical solution of the scheme are proved. For this numerical simulation, we give the energy modulus estimates for intermediate variables and final variables by using the error estimation properties of interpolation operator and L2 projection operator. The results of numerical experiments verify the accuracy of the scheme. Because of the nonlocal property of fractional order operators, the coefficient matrices of linear equations are dense matrices in the discrete schemes. For a problem of order N, the storage of matrix is O (N2), and the computation of direct solution (such as Gauss elimination method) is O (N3), which is accompanied by the increase of N. The complexity of the problem will make the computation time too long and lose the efficiency of the algorithm. Therefore, we need to find an accelerated computing method for solving this kind of problem. When we select piecewise constant polynomial function and piecewise linear polynomial function to approximate intermediate variable and final variable respectively, we find that the coefficient matrix corresponding to discrete scheme has or partly has Toeplitz structure. We know that the storage of Toeplitz matrix can be reduced to O (N), and the computation of Toeplitz matrix-vector product is O (N log N),. Therefore, we can design a fast algorithm for solving this kind of linear equations on the basis of conjugate gradient method. So that the memory of the matrix is O (N), the computation of each iteration is O (N log N). For some matrices with poor condition number, adding a suitable preprocessor can further reduce the number of iterations and improve the computational efficiency. The effectiveness of this fast algorithm is verified by numerical experiments.
【學(xué)位授予單位】：山東師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.82
本文編號：2383567
[Abstract]:In this paper, we consider a class of nonlocal generalized elastic models, where 惟 = (0 ~ 1), d ~ 0 and d ~ 0 denote the left and right diffusivity coefficient, denote the source term, and denote the left and right Riemann-Liouville fractional derivative operators of 尾 order respectively. In this model, there are integral operations and derivation operations at the same time, so it is natural to introduce an intermediate variable to solve this kind of integrodifferential equation. By taking the fractional derivative part of the integrable function as the intermediate variable, we decompose the original problem into an integral equation of order 1- 偽 and a differential equation of order 尾, and then define its mixed form variational scheme. Because the two equivalent equations do not need to satisfy a certain coupling relation and can be solved independently, we only need to prove that the bilinear form is mandatory and continuous in the space H- (1- 偽) / 2 (惟) 脳 H0 尾 / 2 (惟). According to the Lax-Milgram 's Lemma, we can obtain the proper definiteness of the variational decomposition of the mixed problem. In the discussion of the fitness of solutions for fractional integral equations, we also obtain a criterion for the solvability of a class of first type Fredholm integral equations in space H- (1- 偽) / 2 (惟). Based on the variational principle of mixed form, the finite element discrete scheme of mixed form is further defined, and the existence and uniqueness of the numerical solution of the scheme are proved. For this numerical simulation, we give the energy modulus estimates for intermediate variables and final variables by using the error estimation properties of interpolation operator and L2 projection operator. The results of numerical experiments verify the accuracy of the scheme. Because of the nonlocal property of fractional order operators, the coefficient matrices of linear equations are dense matrices in the discrete schemes. For a problem of order N, the storage of matrix is O (N2), and the computation of direct solution (such as Gauss elimination method) is O (N3), which is accompanied by the increase of N. The complexity of the problem will make the computation time too long and lose the efficiency of the algorithm. Therefore, we need to find an accelerated computing method for solving this kind of problem. When we select piecewise constant polynomial function and piecewise linear polynomial function to approximate intermediate variable and final variable respectively, we find that the coefficient matrix corresponding to discrete scheme has or partly has Toeplitz structure. We know that the storage of Toeplitz matrix can be reduced to O (N), and the computation of Toeplitz matrix-vector product is O (N log N),. Therefore, we can design a fast algorithm for solving this kind of linear equations on the basis of conjugate gradient method. So that the memory of the matrix is O (N), the computation of each iteration is O (N log N). For some matrices with poor condition number, adding a suitable preprocessor can further reduce the number of iterations and improve the computational efficiency. The effectiveness of this fast algorithm is verified by numerical experiments.
【學(xué)位授予單位】：山東師范大學(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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