分?jǐn)?shù)階偏微分方程的幾類有限元方法研究
發(fā)布時(shí)間：2019-03-16 15:03
【摘要】：由于分?jǐn)?shù)階導(dǎo)數(shù)能夠比整數(shù)階導(dǎo)數(shù)更準(zhǔn)確地描述具有記憶和遺傳性質(zhì)的材料與物理過(guò)程,分?jǐn)?shù)階微分方程在諸多領(lǐng)域得到了廣泛應(yīng)用和深入研究.但是分?jǐn)?shù)階微分方程的解析解通常很難求出,即便能求出解析解,大多數(shù)的解也都含有無(wú)窮級(jí)數(shù)或一些難以計(jì)算的特殊函數(shù),所以人們更加關(guān)注分?jǐn)?shù)階微分方程的數(shù)值解法.目前在對(duì)分?jǐn)?shù)階微分方程所提出的各種數(shù)值方法中,有限元方法由于區(qū)域適應(yīng)性強(qiáng),網(wǎng)格剖分靈活,并且簡(jiǎn)單通用,對(duì)解的光滑性要求不高,因而更受關(guān)注.本文研究了分?jǐn)?shù)階偏微分方程的幾類有限元方法,其中重點(diǎn)研究了時(shí)間分?jǐn)?shù)階微分方程的降階有限元方法和空間分?jǐn)?shù)階微分方程及方程組的時(shí)空有限元方法.所做的工作可以分為如下三部分.第一部分即本文的第三章研究了時(shí)間分?jǐn)?shù)階Cable方程的標(biāo)準(zhǔn)有限元方法.采用傳統(tǒng)有限元方法的思想,在時(shí)間方向上采用有限差分離散.空間方向上利用有限元近似,得到了分?jǐn)?shù)階Cable方程的Crank-Nicolson型全離散格式.我們對(duì)時(shí)間分?jǐn)?shù)階導(dǎo)數(shù)的差分離散給出了不同于L1算法的新的系數(shù),所得到的全離散格式關(guān)于時(shí)間層數(shù)n具有更好的繼承性.文中還詳細(xì)給出了全離散格式的穩(wěn)定性分析和最優(yōu)階L2模誤差估計(jì)結(jié)果,而且分別展示了空間一維和二維的數(shù)值例子,驗(yàn)證了理論分析的正確性.第二部分利用基于特征投影分解(POD)理論的降階有限元方法分別分析了空間二維的時(shí)間分?jǐn)?shù)階擴(kuò)散方程、時(shí)間分?jǐn)?shù)階Tricomi型方程以及時(shí)間分?jǐn)?shù)階Sobolev方程.所采用的降階方法是：先利用一般有限元格式計(jì)算出在前面很短時(shí)間段內(nèi)的有限元解,將其作為瞬像；然后從瞬像中找到在最小平方意義下的最優(yōu)POD基,這些基函數(shù)的個(gè)數(shù)遠(yuǎn)遠(yuǎn)小于一般有限元方法的基函數(shù)的個(gè)數(shù),將它們張成的空間作為降階有限元空間；最后利用降階有限元格式求得近似解.在時(shí)間分?jǐn)?shù)階微分方程的有限元方法中,要求tn時(shí)的解,需要將前面所有滿足ttn。的時(shí)間層上的解存儲(chǔ)并疊加起來(lái).由于我們?cè)诿總�(gè)時(shí)間層上降低了解空間的自由度,整體的存儲(chǔ)量和計(jì)算量就得到了大幅度降低,從而減輕了分?jǐn)?shù)階微分算子的非局部性帶來(lái)的計(jì)算負(fù)擔(dān).在文中,我們給出了分?jǐn)?shù)階微分方程的全離散有限元格式和降階有限元格式并分別進(jìn)行了穩(wěn)定性和收斂性分析,得到了有限元解和降階有限元解的L2模最優(yōu)階誤差估計(jì)結(jié)果.數(shù)值算例也表明,基于POD理論的降階有限元方法能夠保證在精度上不低于傳統(tǒng)有限元方法的情況下極大地降低了存儲(chǔ)量,提高了計(jì)算效率.本部分的內(nèi)容安排在論文的第四章至第六章,在第五章中我們還具體給出了降階有限元方法的算法流程.本文的第三部分研究了空間分?jǐn)?shù)階微分方程(組)的時(shí)空有限元方法.在第七章中,利用時(shí)間間斷而空間連續(xù)的間斷時(shí)空有限元方法研究了半線性空間分?jǐn)?shù)階擴(kuò)散方程.對(duì)近似變分問(wèn)題,通過(guò)適當(dāng)選取試探函數(shù)得到了可以逐時(shí)間層推進(jìn)求解的時(shí)空有限元全離散格式.在理論分析中,利用Radau積分公式將近似解表示為基于Radau積分點(diǎn)的Lagrange插值多項(xiàng)式的線性組合的形式,從而在不對(duì)時(shí)空網(wǎng)格施加任何限制條件的情況下給出了弱解的存在唯一性證明.通過(guò)引入橢圓投影算子,利用尼采(Nitche)技巧給出了分?jǐn)?shù)階橢圓投影的L2模估計(jì),進(jìn)而詳細(xì)導(dǎo)出了時(shí)空有限元解的最優(yōu)階L∞(L2)模誤差估計(jì)結(jié)果.在第八章,利用時(shí)間間斷的時(shí)空有限元方法構(gòu)造了非線性空間分?jǐn)?shù)階反應(yīng)擴(kuò)散方程組的全離散格式并給出了格式的適定性分析和誤差估計(jì),從而將間斷時(shí)空有限元方法進(jìn)一步推廣到了分?jǐn)?shù)階方程組.理論分析和數(shù)值模擬結(jié)果均顯示,間斷時(shí)空有限元方法能夠在時(shí)間和空間兩個(gè)方向上同時(shí)達(dá)到高階精度,對(duì)空間分?jǐn)?shù)階微分方程(組)仍適用.這為今后研究更復(fù)雜的分?jǐn)?shù)階方程(組)奠定了基礎(chǔ).
[Abstract]:Since the fractional derivative can describe the material and the physical process with the memory and the genetic property more accurately than the integral order derivative, the fractional differential equation has been widely used and studied in many fields. But the analytic solution of fractional differential equation is usually hard to find, even if the analytic solution can be obtained, most of the solutions contain infinite series or some special functions which are difficult to calculate, so the numerical solution of the fractional differential equation is more concerned. At present, in the various numerical methods proposed by fractional differential equations, the finite element method is more and more concerned because of the strong adaptability of the region, the flexibility of the mesh, and the simplicity and general purpose. The finite element method of fractional partial differential equation is studied in this paper. The finite element method of the order differential equation and the space-time finite element method of the space fractional differential equation and the system are studied. The work done can be divided into three parts. In the first part, the third chapter of this paper studies the standard finite element method of the time-fractional Cable equation. With the idea of the traditional finite element method, the finite difference dispersion is used in the time direction. The Crank-Nicolson type full-discrete format of the fractional-order Cable equation is obtained by using the finite element approximation in the spatial direction. We give a new coefficient which is different from that of the L1 algorithm, and the obtained full-discrete format has better inheritance with respect to the time layer n. In this paper, the stability analysis and the optimal order L2 model error estimation result of the full-discrete format are also given in detail, and the numerical examples of the two-dimensional space one and two dimensions are shown, and the correctness of the theoretical analysis is verified. In the second part, the time-fraction-order diffusion equation, the time-fraction-order Triomi-type equation and the time-order Sobolev equation of the two-dimensional space are respectively analyzed by the reduced-order finite element method based on the feature-based projection decomposition (POD) theory. The method comprises the following steps of: firstly, using a general finite element format to calculate a finite element solution in a short time period, The number of these base functions is much smaller than the number of basis functions of the general finite element method, and the space formed by them is taken as the order-order finite element space, and the approximate solution is obtained by using the reduced-order finite element method. In the finite element method of the differential equation of time fractional order, the solution at tn is required, and all the preceding tns are required to be satisfied. The solution on the time layer is stored and superimposed. As we lower the degree of freedom of understanding the space on each time layer, the total storage amount and the calculation amount are greatly reduced, so that the computational burden of the non-locality of the fractional-order differential operator is reduced. In this paper, we present the full-discrete finite element method and the order-order finite element format of the fractional order differential equation, and the stability and the convergence analysis are respectively carried out, and the result of the optimal order error of the L2 mode of the finite element solution and the order-reduction finite element solution is obtained. The numerical example also shows that the reduced-order finite element method based on the POD theory can ensure that the storage capacity is greatly reduced under the condition that the precision is not lower than the traditional finite element method, and the calculation efficiency is improved. The content of this part is in Chapter 4 to Chapter 6 of the thesis. In the fifth chapter, we also give the algorithm flow of the order-reduction finite element method. The third part of this paper studies the space-time finite element method of the spatial fractional differential equation (group). In chapter 7, the semi-linear space fractional diffusion equation is studied by the discontinuous space-time finite element method. For the approximate variational problem, the time-space finite element full-discrete format that can be solved by the time-by-time layer can be obtained by the appropriate selection of the heuristic function. In the theoretical analysis, the approximate solution is represented as a linear combination of the Lagrange interpolation polynomial based on the Radu integral point by the Radu integral formula, so that the existence and uniqueness of the weak solution is proved without applying any restriction condition to the space-time grid. By introducing the elliptic projection operator, the L2 mode estimation of the fractional ellipse projection is given by using the Nitche technique, and the result of the optimal order L-(L2) mode error estimation of the time-space finite element solution is derived in detail. In chapter 8, the time-space finite element method is used to construct the full-discrete format of the nonlinear space fractional-order reaction diffusion system, and the appropriate qualitative and error estimates of the format are given, so that the discontinuous space-time finite element method is further extended to the fractional-order equations. Both the theoretical analysis and the numerical simulation results show that the discontinuous space-time finite element method can achieve high-order precision in both time and space, and the spatial fractional differential equation (group) is still applicable. This provides the basis for studying more complex fractional-order equations (groups) in the future.
【學(xué)位授予單位】：內(nèi)蒙古大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.82
本文編號(hào)：2441607
[Abstract]:Since the fractional derivative can describe the material and the physical process with the memory and the genetic property more accurately than the integral order derivative, the fractional differential equation has been widely used and studied in many fields. But the analytic solution of fractional differential equation is usually hard to find, even if the analytic solution can be obtained, most of the solutions contain infinite series or some special functions which are difficult to calculate, so the numerical solution of the fractional differential equation is more concerned. At present, in the various numerical methods proposed by fractional differential equations, the finite element method is more and more concerned because of the strong adaptability of the region, the flexibility of the mesh, and the simplicity and general purpose. The finite element method of fractional partial differential equation is studied in this paper. The finite element method of the order differential equation and the space-time finite element method of the space fractional differential equation and the system are studied. The work done can be divided into three parts. In the first part, the third chapter of this paper studies the standard finite element method of the time-fractional Cable equation. With the idea of the traditional finite element method, the finite difference dispersion is used in the time direction. The Crank-Nicolson type full-discrete format of the fractional-order Cable equation is obtained by using the finite element approximation in the spatial direction. We give a new coefficient which is different from that of the L1 algorithm, and the obtained full-discrete format has better inheritance with respect to the time layer n. In this paper, the stability analysis and the optimal order L2 model error estimation result of the full-discrete format are also given in detail, and the numerical examples of the two-dimensional space one and two dimensions are shown, and the correctness of the theoretical analysis is verified. In the second part, the time-fraction-order diffusion equation, the time-fraction-order Triomi-type equation and the time-order Sobolev equation of the two-dimensional space are respectively analyzed by the reduced-order finite element method based on the feature-based projection decomposition (POD) theory. The method comprises the following steps of: firstly, using a general finite element format to calculate a finite element solution in a short time period, The number of these base functions is much smaller than the number of basis functions of the general finite element method, and the space formed by them is taken as the order-order finite element space, and the approximate solution is obtained by using the reduced-order finite element method. In the finite element method of the differential equation of time fractional order, the solution at tn is required, and all the preceding tns are required to be satisfied. The solution on the time layer is stored and superimposed. As we lower the degree of freedom of understanding the space on each time layer, the total storage amount and the calculation amount are greatly reduced, so that the computational burden of the non-locality of the fractional-order differential operator is reduced. In this paper, we present the full-discrete finite element method and the order-order finite element format of the fractional order differential equation, and the stability and the convergence analysis are respectively carried out, and the result of the optimal order error of the L2 mode of the finite element solution and the order-reduction finite element solution is obtained. The numerical example also shows that the reduced-order finite element method based on the POD theory can ensure that the storage capacity is greatly reduced under the condition that the precision is not lower than the traditional finite element method, and the calculation efficiency is improved. The content of this part is in Chapter 4 to Chapter 6 of the thesis. In the fifth chapter, we also give the algorithm flow of the order-reduction finite element method. The third part of this paper studies the space-time finite element method of the spatial fractional differential equation (group). In chapter 7, the semi-linear space fractional diffusion equation is studied by the discontinuous space-time finite element method. For the approximate variational problem, the time-space finite element full-discrete format that can be solved by the time-by-time layer can be obtained by the appropriate selection of the heuristic function. In the theoretical analysis, the approximate solution is represented as a linear combination of the Lagrange interpolation polynomial based on the Radu integral point by the Radu integral formula, so that the existence and uniqueness of the weak solution is proved without applying any restriction condition to the space-time grid. By introducing the elliptic projection operator, the L2 mode estimation of the fractional ellipse projection is given by using the Nitche technique, and the result of the optimal order L-(L2) mode error estimation of the time-space finite element solution is derived in detail. In chapter 8, the time-space finite element method is used to construct the full-discrete format of the nonlinear space fractional-order reaction diffusion system, and the appropriate qualitative and error estimates of the format are given, so that the discontinuous space-time finite element method is further extended to the fractional-order equations. Both the theoretical analysis and the numerical simulation results show that the discontinuous space-time finite element method can achieve high-order precision in both time and space, and the spatial fractional differential equation (group) is still applicable. This provides the basis for studying more complex fractional-order equations (groups) in the future.
【學(xué)位授予單位】：內(nèi)蒙古大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.82
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