Journal of Scientific Computing ( IF 2.8 ) Pub Date : 2020-05-30 , DOI: 10.1007/s10915-020-01230-z Samir Karaa
We derive optimal \(L^2\)-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order \(\alpha \in (0,1)\), for cases with smooth and nonsmooth initial data. A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods. The analysis is based on a semigroup type approach and exploits the properties of the inverse of the associated elliptic operator. Completely discrete schemes are analyzed in the same framework using a backward Euler convolution quadrature method in time. Numerical examples including conforming, nonconforming and mixed finite element methods are presented to illustrate the theoretical results.
中文翻译:
半线性时间分数维扩散问题的Galerkin型方法
对于具有平滑和非平滑初始数据的情况,我们针对涉及Caputo导数的半线性时间分数次扩散问题,以时间\(\ alpha \ in(0,1)\)的时间得出最优\(L ^ 2 \) -误差估计。介绍了一个通用框架,该框架允许对Galerkin类型空间逼近方法进行统一的误差分析。该分析基于半群类型方法,并利用了相关的椭圆算子的逆的性质。在同一框架中,使用后向Euler卷积正交方法及时分析了完全离散的方案。数值算例包括相容,非相容和混合有限元方法,以说明理论结果。