非规则二维区域上空间分数阶扩散方程的有限元方法

Fractional differential equations have been widely used in numerical simulation of computational science and engineering. Riesz and Riemann-Liouville (R-L) space fractional diffusion equations are important fractional differential equations. So far, finite element methods devised for these two kinds of equations are limited to one-dimensional domains or regular two-dimensional (2D) ones.. This project is focused on Riesz and Riemann-Liouville (R-L) space fractional diffusion equations in irregular domains. First we will study the effective finite element method to reduce time-consuming for obtaining stiffness matrix and minimize complexity of stiffness matrix structure which are derived from the nonlocal property of Riesz and R-L fractional operators. Then we will discuss the transformation between Riesz fractional diffusion equation and the integer-order differential equations and develop finite element method for the obtained equation. Last but not least, we will consider the multigrid method for the obtained systems of algebraic equations from the finite element methods. There are several difficulties such as construction of new finite element basis functions for fractional problems, suitable transformation method for fractional equations, design of the multigrid methods and relevant convergence analysis involved in this research. Thus some novel approaches are needed.. The project aims to develop the theory of finite element method for space fractional diffusion equations, provide an efficient algorithm for fractional diffusion equations on irregular 2D domain and be applied to solve a number of practical problems.

分数阶微分方程广泛应用于科学工程的数值计算与模拟，Riesz和Riemann-Liouville(R-L)空间分数阶扩散方程是其中重要的两类，目前关于这两类方程有限元方法的研究仅限于一维和二维(2D)规则区域。. 本项目针对非规则2D区域上Riesz和R-L空间分数阶扩散方程，首先研究在有限元方法构造过程中由Riesz和R-L分数阶算子非局部性导致刚度矩阵形成耗时大、结构复杂的问题；接着研究Riesz分数阶扩散方程与整数阶微分方程之间的转化问题，以及相应的有限元方法；最后针对上述有限元代数方程组，研究其多重网格方法。这些研究涉及分数阶方程有限元新基函数构造、合适的转化方法、多重网格法的设计与收敛性分析等难点，需要发展新方法、新技术和新技巧。. 本项目旨在发展分数阶扩散方程有限元求解的相关理论，为非规则2D区域上分数阶扩散方程提供高效算法，并应用于一些实际问题的求解。

本项目主要研究非规则2D区域上空间分数阶扩散方程的有限元高效算法。我们主要研究了空间分数阶扩散方程新有限元基函数的构造，大大降低了有限元刚度矩阵的生成耗时。我们针对2D时空分数阶Caputo-Riesz分数阶扩散方程构造了高效的有限差分/有限元方法、时空有限元方法，并分析了数值格式的稳定性和收敛性。对分数阶扩散方程利用有限元离散得到的代数系统，我们设计了低复杂度、低存储的多重网格方法。我们也研究了时间分数阶扩散方程的数值求解，构造了时间方向高阶收敛且低存储的数值方法。同时，我们也将所得成果应用于生物医学等领域数学模型的求解。目前，我们已经发表SCI论文13篇。