Efficient Numerical Methods And Applications Of Several Fractional Partial Differential Equations And Peridynamic Models

Classical integer-order diffusion equation was first proposed by Fick during his study of the propagation of nutrients within the membranes of living organ-isms,and then derived by Einstein and Pearson based on the first principle and the random walk,respectively.There are two common assumptions in their orig-inal derivations,i.e.,(i)the existence of a mean free path and(ii)the existence of a mean waiting time in the underlying particle movements.Based on these assumptions,the movement of the particles obeys the classic Gauss distribution,and the Fokker-Planck equation is the classical diffusion equation.But many experiments show that these two assumptions are not satisfied for the inhomoge-neous medium,the movement of the particles exhibits heavy-tailed waiting time,long-range spatial interaction or strong asymmetric plume.Since the fundamental solutions to classical integer-order diffusion equation are exponentially decaying,thus it cannot accurately describe these anomalous diffusion phenomena.The solutions to fractional diffusion equations are power law decaying,thus they pro-vide more accurate descriptions on these challenging phenomena.Anomalous diffusion phenomena exist widely in nature and various engineering fields,such as viscoelastic fluid,biological blockage in rivers and groundwater and the long tail plume during the sediment transport in estuary.Therefore,the mathematics and numerical studies on fractional partial differential equations(FPDEs)have strong theoretical and practical significances,and have become a popular research field.In solid mechanics,the classical PDEs model assume that all of the internal forces are local.However,because of its differentiability assumptions on the displacement field,the PDEs model can not accurately describe the discontinuous problems,such as the damage evolution of materials and fracture damage.Silling[98]developed the peridynamic(PD)theory to overcome this difficulty,whose basic idea is to replace the original PDEs model by nonlocal integral model.In the PD theory,the constitutive model relies on the finite deformation vector rather than the deformation gradient,thus it is particular suitable to describe the problems with discontinuous displacement field in the solid mechanics.Compared to classical integer-order PDEs,both of the FPDEs and the PD model exhibit nonlocal features,numerical discretizations of these models give rise to dense or full coefficient matrices or historical dependence.We use the Riesz space-fractional diffusion equation to expose the idea,the coefficient matrix arising from its numerical discretizaiton is full,which can be stored in(?)(N2)memory with N being the number of spatial unknowns.At each time step,direct solvers such as the Gauss elimination can be used to inverse the coefficient matrix with the computational cost of(?)(N3).The Krylov subspace iteration methods can be used to solve the linear system with the computational complexity of(?)(N2)per iteration,but the memory requirements are still(?)(N2).These difficulties are never encountered when dealing with integer-order PDEs,and especially for large scale or high dimensional problems.This thesis mainly study the efficient numerical methods for FPDEs and related nonlocal models with applications to phase field simulations and fracture mechanics.The structure of this thesis are organized as follows:In chapter 1,we firstly outline the fractional operators and PD theory,and then we recall two special matrices,including the Toeplitz matrix and the circulant matrix.In chapter 2,we develop a series of fast discontinuous Galerkin(DG)finite element methods for a bond-based linear PD model in one space dimension.More precisely,we develop a preconditioned fast piecewise-constant DG scheme on a geometrically graded locally refined composite mesh to handle the scenario that the jump discontinuity of the displacement field occurs at one of the nodes in the original uniform partition.We also develop a preconditioned fast piecewise-linear DG scheme on a uniform mesh that has a second-order convergence rate when the jump discontinuity occurs at one of the computational nodes or has a convergence rate of one-half order otherwise.Motivated by these results,we develop a preconditioned fast hybrid DG scheme that is discretized on a locally uniformly refined composite mesh to numerically simulate the PD model where the jump discontinuity of the displacement field occurs inside a computational cell.We use a piecewise-constant DG scheme on the uniform mesh of mesh size h and a piecewise-linear DG scheme on the locally uniformly refined mesh of mesh size h2,so that the hybraid DG scheme has an overall convergence rate of first-order.Numerical experiments show the accuracy and efficiency of the proposed schemes.In chapter 3,we study time-fractional Allen-Cahn and Cahn-Hilliard phase-field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials.Due to the nonlocal behavior of the time-fractional operators,at each time step,the informations in all previous time steps are needed which leads to an overall com-putational complexity of(?)(M2N)and memory requirements of(?)(MN),where M,N are the number of time steps and spatial unknowns.We utilize the sum-of-exponentials(SOE)[65]approximation to develop efficient finite difference scheme and Fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity.Since the proof of the energy dissipation law is still an open problem,thus it is worthy to numerically study the energy decay behavior of the time-fractional phase-field models.For the time-fractional Cahn-Hilliard equation,we observe that the bigger the fractional order a,the faster the energy decays.This conclu-sion is physically reasonable,since the time-fractional operator is corresponding to the sub-diffusion process.However,we obtained an opposite conclusion for the time-fracitonal Allen-Cahn equation.Moreover,we also study the coarsening dynamics of the time-fractional Cahn-Hilliard equation,numerical results reveal that the scaling law for the energy decays as(?)(t-?/3),which is consistent with the well-known result(?)(t-1/3)of the classical Cahn-Hilliard equation.In chapter 4,we investigate the finite element approximation of a space-time fractional Allen-Cahn equation,where the classical Laplacian operator is replaced by the integral fractional Laplacian.Due to the nonlocal features of frac-tional derivatives,conventional solvers require(?)(N2+MN)memory units and(?)(M2N+MN3)computational complexity.By exploring the special structure of the stiffness matrix and utilizing the divide-and-conquer strategy,we developed an efficient linearized divide-and-conquer Galerkin finite element method with-out resorting to any lossy compression.This new method reduces the memory requirement to(?)(MN)and the operation count to(?)(MN(log2 M+log N)).We employ the "Invariant Energy Quadratization" approach to alleviate the time step constraint.Numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space-time fractional Allen-Cahn equation.In chapter 5,we develop a Crank-Nicolson alternating direction implicit finite volume(CN-ADI-FV)method for time-dependent Riesz space-fractional diffusion equation in two space dimensions.Norm-based stability and convergence analy-sis are given to show that the developed method is unconditional stable and of second-order accuracy both in space and time.Compared with the conventional finite volume scheme,the developed CN-ADI-FV scheme reduces the computa-tional complexity of original finite volume scheme from(?)(N3)=(?)(Nx3Ny3)to(?)(Nx3Ny+NxNy3),where Nx and Ny are the number of partitions in x and y directions,respectively.By using the permutation matrix,we develop an ef-ficient lossless CN-ADI-FV scheme which only has(?)(N)memory requirement and(?)(N log N)computational complexity per iteration.Several numerical ex-periments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large-scale modeling and simulations.