多项时间分数阶扩散方程各向异性线性三角元的高精度分析

在各向异性网格下,针对具有Caputo导数的二维多项时间分数阶扩散方程,给出了线性三角形元的高精度分析.首先,基于线性三角形元和改进的L1格式,建立了一个全离散逼近格式,并证明了其无条件稳定性;其次,利用有限元插值算子与Riesz投影算子之间的关系及相关的高精度结果,导出了超逼近性质.进而,借助于插值后处理技术得到了超收敛估计.值得指出的是,单独利用插值算子或Riesz投影都无法得到上述超逼近和超收敛结果.最后,利用数值算例验证了理论分析的正确性.此外,对一些常见的有限单元在该方程的数值逼近方面,作了进一步探讨.

HIGH ACCURACY ANALYSIS OF ANISOTROPIC LINEAR TRIANGULAR ELEMENT FOR MULTI-TERM TIME FRACTIONAL DIFFUSION EQUATIONS

High accuracy analysis of linear triangular element is proposed for two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivative on anisotropic meshes. Firstly, based on linear triangular element and modified L1 scheme, a fully-discrete approximate scheme is established and the unconditional stability analysis is investigated. Secondly, by use of the relationship between the interpolation operator and Riesz projection operator, superclose property is derived by related known high accuracy results. Moreover, the superconvergence estimate is obtained through the interpolation postprocessing technique. It is worth mentioning that the above superclose and superconvergence results will not be derived by the interpolation operator and Riesz projection operator alone. Finally, numerical results are provided to confirm the validity of our theoretical analysis. Furthermore, some popular finite elements of numerical approximation for the focused equation are investigated.