基于分层矩阵存储的分数阶扩散方程的自适应算法设计和应用

Fractional Differential Equation is a class of equation involving fractional derivatives, which is defined by integral differential operators. Thanks to the wide utilizations in the fields of hydrology, physics, materials science, etc., in recent years the theory, applications and numerical methods for fractional differential equations attracted researchers all over the world. Due to the nonlocality of the fractional operator, the current applications of the finite element method, finite difference method, spectral methods and some other numerical methods for solving fractional differential equations, result in full dense matrices, which brings in huge challenges in computation and memory storage. Therefore, reducing the amount of memory storage in numerical calculation and the computation time, to make the application to time-sensitive or large-scale practical problems possible.This project devotes to solve fractional diffusion equations, using framework of finite element method based on Hierarchical matrix, then established multigrid method on this basis, to construct less memory, less computation algorithms. The current project will use the Taylor expansions of the weak singular kernel, low rank matrix approximation of sub-block matrix, the set of admissible conditions and some other methods to carry out the design of efficient algorithms. At the same time, taking advantage of the adaptive method, to refine the mesh for the singular region of solutions automatically, eventually we obtain the optimal convergence rate overall the calculation region.

分数阶微分方程中所包含的分数阶导数是由积分微分算子定义的非局部算子。 因其在水文学，物理学，材料学等科学领域的广泛应用，近年来关于分数阶微分方程的理论、应用及数值方法的研究备受国内外学者的关注。 受分数阶导数算子非局部性的影响，目前应用有限元方法、有限差分方法、谱方法等数值方法求解分数阶微分方程时，所得离散矩阵都是稠密的，这给计算和存储带来了很大的挑战。 因此，在数值计算时减少存储量和降低计算耗时，才能使数值方法应用于具有时效性或规模较大实际问题成为可能。 本项目通过基于分层矩阵方法的有限元离散方法，对分数阶扩散方程建立多重网格快速计算框架，从而构造存储量少、计算量小的算法。 本项目将采用算子弱奇异核的泰勒展开、矩阵子块的低秩逼近和网格剖分的容许条件设定等方法来开展高效算法的设计。 同时，利用自适应方法，对解的奇异性区域进行网格加密，以获得算法在计算区域上整体最优精度。