Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation
Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation
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DOI：
10.1137/13093933x
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发表时间：
2014-01
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通讯作者：
Bangti Jin;R. Lazarov;J. Pasciak;Zhi Zhou
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文献类型：
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作者：
Bangti Jin;R. Lazarov;J. Pasciak;Zhi Zhou
考虑一类带Riemann-Liouville型空间分数阶导数的一维分数阶抛物型方程的初边值问题，其阶为$\alpha\in（1，2）$.我们研究了空间半离散计划，使用标准的Galerkin有限元法与分段线性有限元，以及完全离散计划的基础上向后欧拉方法和曲柄-尼科尔森方法。半离散格式的误差估计在$L^2（D）$-和$H^{\alpha/2}（D）$-范数下得到，全离散格式的误差估计在$L^2（D）$-范数下得到.这些估计涵盖了光滑和非光滑的初始数据，并直接表示在初始数据的平滑度。大量的数值结果来说明理论结果。
We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann--Liouville type and order $\alpha\in (1,2)$. We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank--Nicolson method. Error estimates in the $L^2(D)$- and $H^{\alpha/2}(D)$-norm are derived for the semidiscrete scheme and in the $L^2(D)$-norm for the fully discrete schemes. These estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.