Inhomogeneous Dirichlet Boundary-Value Problems of Space-Fractional Diffusion Equations and their Finite Element Approximations
Inhomogeneous Dirichlet Boundary-Value Problems of Space-Fractional Diffusion Equations and their Finite Element Approximations
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DOI：
10.1137/130932776
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发表时间：
2014-06
期刊：
影响因子：
--
通讯作者：
Hong Wang;Danping Yang;Shengfeng Zhu
中科院分区：
文献类型：
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作者：
Hong Wang;Danping Yang;Shengfeng Zhu
证明了常系数或变系数保守Caputo空间分数扩散方程的非齐次Dirichlet边值问题的Galerkin弱公式和Petrov—Galerkin弱公式的适定性。我们还证明了它们的黎曼-刘维尔类似物的弱解一般不存在。此外，我们发展了一种间接有限元法求解Caputo分数阶微分方程的Dirichlet边值问题，使变系数分数阶扩散方程的数值解的计算量从$O(N^3)$减少到$O(N)$，并使任意准均匀空间分区上的内存需求从$O(N^2)$减少到$O(N)$。我们进一步证明了该方法的近似尖锐误差估计，该估计仅用问题的规定数据的平滑性来表示。通过数值实验，比较了该方法与Galerkin有限元法的性能。
We prove the wellposedness of the Galerkin weak formulation and Petrov--Galerkin weak formulation for inhomogeneous Dirichlet boundary-value problems of constant- or variable-coefficient conservative Caputo space-fractional diffusion equations. We also show that the weak solutions to their Riemann--Liouville analogues do not exist, in general. In addition, we develop an indirect finite element method for the Dirichlet boundary-value problems of Caputo fractional differential equations, which reduces the computational work for the numerical solution of variable-coefficient fractional diffusion equations from $O(N^3)$ to $O(N)$ and the memory requirement from $O(N^2)$ to $O(N)$ on any quasiuniform space partition. We further prove a nearly sharp error estimate for the method, which is expressed in terms of the smoothness of the prescribed data of the problem only. We carry out numerical experiments to investigate the performance of the method in comparison with the Galerkin finite element method.