In contrast with the diffusion equation which smoothens the initial data to $C^\infty$ for $t>0$ (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac--Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.

翻译：与扩散方程在$t>0$时（远离区域边角/边缘）将初始数据光滑至$C^\infty$不同，次扩散方程仅表现出有限的空间正则性。因此，在求解非光滑初始数据的次扩散方程时，通常无法期望获得空间高阶精度。本文针对非光滑初始数据的次扩散方程，构建了一种新的解分裂方法用于高阶有限元逼近。该方法通过将解分裂为两部分（即与时间相关的光滑部分和与时间无关的非光滑部分），并采用不同策略分别逼近这两部分。与时间相关的光滑部分采用空间高阶有限元法和时间卷积求积法逼近，而稳态非光滑部分可通过采用更细网格尺寸或其他可产生高阶精度的方法进行逼近。本文给出多个算例展示如何精确逼近稳态非光滑部分，包括分片光滑初始数据、狄拉克-δ点初始数据以及集中在界面上的狄拉克测度。该论证可直接推广至具有非光滑源项的次扩散方程。大量数值实验支持理论分析，并验证了所提出高阶分裂有限元方法的性能表现。