分数阶偏微分方程的几类有限元方法研究
发布时间：2019-03-16 15:03
【摘要】：由于分数阶导数能够比整数阶导数更准确地描述具有记忆和遗传性质的材料与物理过程,分数阶微分方程在诸多领域得到了广泛应用和深入研究.但是分数阶微分方程的解析解通常很难求出,即便能求出解析解,大多数的解也都含有无穷级数或一些难以计算的特殊函数,所以人们更加关注分数阶微分方程的数值解法.目前在对分数阶微分方程所提出的各种数值方法中,有限元方法由于区域适应性强,网格剖分灵活,并且简单通用,对解的光滑性要求不高,因而更受关注.本文研究了分数阶偏微分方程的几类有限元方法,其中重点研究了时间分数阶微分方程的降阶有限元方法和空间分数阶微分方程及方程组的时空有限元方法.所做的工作可以分为如下三部分.第一部分即本文的第三章研究了时间分数阶Cable方程的标准有限元方法.采用传统有限元方法的思想,在时间方向上采用有限差分离散.空间方向上利用有限元近似,得到了分数阶Cable方程的Crank-Nicolson型全离散格式.我们对时间分数阶导数的差分离散给出了不同于L1算法的新的系数,所得到的全离散格式关于时间层数n具有更好的继承性.文中还详细给出了全离散格式的稳定性分析和最优阶L2模误差估计结果,而且分别展示了空间一维和二维的数值例子,验证了理论分析的正确性.第二部分利用基于特征投影分解(POD)理论的降阶有限元方法分别分析了空间二维的时间分数阶扩散方程、时间分数阶Tricomi型方程以及时间分数阶Sobolev方程.所采用的降阶方法是：先利用一般有限元格式计算出在前面很短时间段内的有限元解,将其作为瞬像；然后从瞬像中找到在最小平方意义下的最优POD基,这些基函数的个数远远小于一般有限元方法的基函数的个数,将它们张成的空间作为降阶有限元空间；最后利用降阶有限元格式求得近似解.在时间分数阶微分方程的有限元方法中,要求tn时的解,需要将前面所有满足ttn。的时间层上的解存储并叠加起来.由于我们在每个时间层上降低了解空间的自由度,整体的存储量和计算量就得到了大幅度降低,从而减轻了分数阶微分算子的非局部性带来的计算负担.在文中,我们给出了分数阶微分方程的全离散有限元格式和降阶有限元格式并分别进行了稳定性和收敛性分析,得到了有限元解和降阶有限元解的L2模最优阶误差估计结果.数值算例也表明,基于POD理论的降阶有限元方法能够保证在精度上不低于传统有限元方法的情况下极大地降低了存储量,提高了计算效率.本部分的内容安排在论文的第四章至第六章,在第五章中我们还具体给出了降阶有限元方法的算法流程.本文的第三部分研究了空间分数阶微分方程(组)的时空有限元方法.在第七章中,利用时间间断而空间连续的间断时空有限元方法研究了半线性空间分数阶扩散方程.对近似变分问题,通过适当选取试探函数得到了可以逐时间层推进求解的时空有限元全离散格式.在理论分析中,利用Radau积分公式将近似解表示为基于Radau积分点的Lagrange插值多项式的线性组合的形式,从而在不对时空网格施加任何限制条件的情况下给出了弱解的存在唯一性证明.通过引入椭圆投影算子,利用尼采(Nitche)技巧给出了分数阶椭圆投影的L2模估计,进而详细导出了时空有限元解的最优阶L∞(L2)模误差估计结果.在第八章,利用时间间断的时空有限元方法构造了非线性空间分数阶反应扩散方程组的全离散格式并给出了格式的适定性分析和误差估计,从而将间断时空有限元方法进一步推广到了分数阶方程组.理论分析和数值模拟结果均显示,间断时空有限元方法能够在时间和空间两个方向上同时达到高阶精度,对空间分数阶微分方程(组)仍适用.这为今后研究更复杂的分数阶方程(组)奠定了基础.
[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.
【学位授予单位】：内蒙古大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82
本文编号：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.
【学位授予单位】：内蒙古大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82
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