| The fractional calculus has a long history, which was firstly proposed in the letter L'Hospital wrote to Leibniz in the September of 1695. During the 300 years after that, because the fractional calculus had not gained ex-tensive attention and applications in Physics and Mechanics, the fractional calculus was only a pure theoretical problem researched by many famous mathematical researchers including Euler, Lacroix, Abel, Liouville, Riemann and so on. However, With the deepening of the understanding of complex physical phenomena and the improvement of computer simulation ability, the fractional derivative modelling of the mechanics and engineering problems is more and more important. In particular, in recent years more and more diffusion processes were found to be non-Fickian, which is called anoma-lous diffusion. Because the anomalous diffusion processes have the property of history-dependence and nonlocality which can be properly modeled by fractional derivatives. Therefore, compared with the integer-order dynam-ic equations, the fractional dynamic equations can provide a more efficient description for the complex systems([18,50,57,5]).The new fractional dynamic equations bring new challenges to the math-ematical researchers on both theoretical analysis and numerical computing. With respect to the numerical computing, many numerical methods includ-ing the finite difference method([14,42]), the finite volume method([91]),the finite element method([60,25]) and the spectral method([35,94]) have been proposed to solve the fractional dynamic equations. These methods have been widely used in the numerical simulation of the anomalous diffusion pro-cess. However, the error analysis for these methods have strong regularity assumption. In the Section 2, we construct a counterexample from which we can conclude that even though the coefficients and the right-hand term have sufficient smoothness, we cannot guarantee the true solution has desire regu-larity. This property is quite different with that in the integer-order case(For integer-order dynamic equation, from the regularity theory, we have that the smoothness of the coefficient and the right-hand term can guarantee the s-moothness of the true solution). Therefore, the full regularity assumption for the above methods lacks theoretical support. Moreover, because the true solution does not have enough regularity, the high-accuracy methods such as the high-order finite difference methods, the high-order finite element meth-ods and the standard spectral methods do not perform well even though the data of the fractional dynamic equations have enough regularity.In addition, with respect to the computational efficiency, the coefficient matrix resulted from the discretization of the fractional dynamic equation is usually full matrix. If the scale of the matrix is N, the computer memory of storing the matrix is O(N2). Moreover, if we solve the linear system by the widely used direct method, the computational complexity is O(N3). Hence the simulation for the fractional dynamic systems is always time-consuming while solving multidimensional or large-scale fractional dynamic problems. With respect to this problem, the mathematical expert Hong Wang carefully analyze the structure of the coefficient matrix. Again by the fast Fourier transform, the computer memory can be reduced to O(N) and the compu-tational complexity of solving the corresponding linear system is O(N log N) per Krylov subspace iteration([79,80]).The failure of solid materials and structures is a classical problem in the study of mechanics, and it is also the focus of attention in the fields of machine, aerospace, civil engineering, water conservancy and chemical engi-neering. Before the foundation of the dynamic theory, with the development of computer software and hardware level as well as the improvement of the fracture mechanics and damage mechanics, researchers have proposed vari-ous mechanical model and numerical methods to simulate the solid material and progressive failure process. These models are based on the assumption that all the internal forces of the medium are contact forces, and the final control equations are mostly described by partial differential equations. The traditional finite element method and finite difference method are also based on the assumption of continuous medium. In the simulation process, it is necessary to know the location and size of the fracture, which is difficult to realize in many practical applications. In addition, with the development of fracture, the grid system for the traditional finite element method or fi-nite difference method must be redivided, which has a strong dependence on the mesh grid([33]). Along with the development of discontinuous finite element method, some progress has been made on the fracture and the frac-ture development of solid material, but there are still some limitations in the simulation of multi-dimensional complex fault system.In order to overcome the basic contradiction of the continuum mechanics assumption and the solid material discontinuity. In 2000, based on the non-local effect modeling, Silling first proposed the dynamic model, which is an integral equation([68]). This model does not based on the continuum assumption and the solutions of differential equations, but the solid is viewed to be composed of the material points that contains all material information with the quality. The interaction exists between the material point and the material point and with the increase of the distance between point and point, the force is weakened, so people usually select the ? neighborhood of a point as its influence domain. Under the framework of the theory, the discontinuous phenomena occur naturally and at the same time this theory breaks through the limitations of molecular dynamics in the computational scale, and can show high accuracy in the macro and micro scales.After the approach is put forward, many numerical methods, such as meshless method([64,63,70]), finite element method([15]) and the finite dif-ference method based on the integral([77]), are proposed to solve the dynamic models. In the case of finite element, the numerical solution has been proved to satisfy the optimal error estimation. However, these methods have a com-mon property, especially when the multidimensional problem is solved, the coefficient matrix is a dense matrix or a full matrix(This depends on the size of the horizon ?). Therefore, similar to the fractional order dynamic equation, the coefficient matrix has a storage requirement of O(N2), and the compu-tational complexity of solving the final linear system is O(N3) by the widely used direct methods. In addition, if we use the finite element method to solve the pcridynamic model, the evaluation of each coefficient matrix entry requires 2dintegrals, where dis the dimension. Especially when the peridy-namic model contains singular integral kernel, the numerical computation of the integral is very time-consuming. Similar to the fractional equation, Hong Wang successfully reduced the storage requirement to O(N) and the computations of solving the corresponding linear system to O(NlogN) per Krylov subspace iteration([81,82]).Based on the above considerations, we developed the high-accuracy pre-serving spectral Galerkin method for the space-fractional diffusion equations and the fast collocation methods for the peridynamic models. The arrange-ment of this dissertation is as follows:In the first chapter, we present some basic concepts and properties which will be used in the following chapters. We first give the definitions of Riemann-Liouville fractional derivative and Caputo fractional derivative, Riemann-Liouville fractional and some properties related to these two deriva-tives. Moreover, we present the definitions of some special matrices related to the fast methods.In the second chapter, we present the high-preserving spectral Galerkin method for the fractional diffusion equations, which can guarantee the high-accuracy of the numerical solution, even when the true solution lacks of the required regularity. This method only assumes the coefficient and the right-hand term have the desired regularity. When the true solution of the frac-tional diffusion equation is not smooth enough, the accuracy of the numerical solution for the standard spectral Galerkin method is low. We also proved the error estimation in this chapter. The numerical experiments presented in this chapter show the utility of this method.In the three chapter we proposed the fast collocation method for the two-dimensional peridynamic models. In this chapter, we carefully analyze the structure of the coefficient matrix resulted from the collocation discretiza-tion. We can obtain that the multiplication of the coefficient matrix and any vectors can be computed by 3 multiplications of the block-Toeplitz-Toeplitz-block(BTTB) matrices and the corresponding vectors which will be given in this chapter. Hence, we can reduce the storage requirement from O(N2) to O(N) and the computational work for the corresponding linear system from O(N2) to O(N log N) per Krylov subspace iteration. The numerical experiments show the utility of this method.In the fourth chapter, we proposed two preconditioners for the fast col-location method for the peridynamic models with strong singular integral kernels. The first preconditioner is block-Circulant-Toeplitz-block(BCTB) type. The other one is block-Circulant-Circulant-block(BCCB) type. The numerical experiments in this chapter show that this two preconditioners are both efficient to reduce the iterative number in the Krylov subspace itera-tive method. Moreover, the second preconditioner is more time-saving than the first one, because the inverse of the BCCB type preconditioner is more efficient.In the fifth chapter, we proposed a fast collocation method for the non-local diffusion model in any convex domain using the penalization method which extends the problem in any convex domain to the problem in a rect-angular domain. The physical convex domain belongs to the rectangular domain. Then the coefficient matrix resulting from the collocation approxi-mation can be decomposed to a BTTB matrix and a diagonal matrix. Using this structure of the coefficient matrix, we can reduced the computer storage from O(N2) to O(N) and the computations in each Krylov subspace iteration from O(N2) to O(N log N). The numerical experiments shows the utility of this method. |
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