Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) are studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having variable (space-time) coefficients. Non-uniform IMEX-L1-FEM is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. A discrete fractional Gr\"{o}nwall inequality is proposed, which enables us to derive optimal error estimates in $L^2$- and $H^1$-norms. Numerical experiments are presented to validate our theoretical findings.
翻译:对非统一的隐含L1有限要素法(IMEX-L1-FEM)的稳定性和最佳趋同性分析进行了研究,以研究具有可变(空间-时间)系数的非自联的椭圆形部分的一组时间差线性线性部分差分/内分式方程式。非统一的IMEX-L1-FEM是结合关于时间方向分级网格的IMEX-L1方法以及空间方向的有限要素法进行的。提议采用离散的分数格罗尔({o}墙上的不平等,这使我们能够得出以2美元和1美元诺姆为单位的最佳误差估计数。提出了数值实验,以证实我们的理论结论。