Finite Element Method For Space Fractional Partial Differential Equations On Irregular Domains

Over the rapid development of the past half-century,fractional calculus has been a branch of mathematics with rich content.The non-local property of fractional operators brings advantages in modeling physical problems as well as difficulties in solving fractional differential equations(FDEs)by numerical methods.Till to now,there is plenty of literature on numerical methods for FDEs.However,most of them focus on problems on regular domains,instead of on irregular domains.The finite element method(FEM)is very suitable for solving problems on irregular domains.The main difficulty in solving problems by FEM is the computation of the stiffness matrix.By these considerations,we focus on FEM for FDEs on irregular domains in high dimensional space.The main contents and results are listed below.Firstly,the FEM for FDEs on irregular two-dimensional domains is studied.It is hard to compute the fractional stiffness matrix on unstructured meshes,especially the computation of the value of fractional derivatives.The key point in calculating the fractional derivatives of the basis functions is seeking the integral path for the derivatives.Based on the concept of the influence domain,we devise an algorithm on searching the path to improve the effectiveness of the algorithm on calculating the matrix.Besides,we analyze the stability and convergence of the FEM for non-linear fractional partial differential equations with Riesz fractional derivatives.The numerical results are consistent with our theoretical results.Secondly,considering the lack of literature on FEM for three-dimensional space fractional problems,we expect to fill this gap.Different from the two-dimensional case,we developed a new algorithm on searching integral paths based on the ray-simplex intersection algorithm.Furthermore,the concept of “template matrix” is used to improve the performance of the summation of the sparse matrix in MATLAB.Then we consider how to solve the three-dimensional time-space fractional differential equation.By using Alikhanov formula and FEM,we develop a fully discrete scheme.The theoretical analysis of the scheme shows its efficiency.Besides,we solve the space-time fractional Bloch-Torrey equation on three-dimensional irregular domains for the first time by the developed scheme.Thirdly,we discuss high order finite element method for two-dimensional space-time fractional partial differential equations.To handle the initial singularity of the solution,Alikhanov formula on non-uniform grids is used to approximate the Caputo fractional derivatives.Same as before,we will examine the stability and convergence of the numerical scheme.Considering the difference of the basis function as before,we give the details on calculating the fractional derivatives of them.Also,the fast algorithm of the Alikhanov formula is presented to improve the performance of the numerical scheme.Finally,two technics on improving the accuracy of the numerical method for FDEs are presented.The first one is using Gauss-Jacobi quadrature in calculating the stiffness matrix for FEM,which has never been considered before.Numerical experiments show that the GaussJacobi quadrature can improve the accuracy of the solution for high order FEM significantly.The second one is using a smoothing transform in solving equations with weak singular solutions.By transforming the equation into an equation with a higher regularity solution,we can solve the new equation using the methods which are less effective for the original equation.Numerical examples verified the effectiveness of the smoothing transform.