The solutions with weakly initial singularity arises in a wide variety of equations, for example,  diffusion and subdiffusion equations. When the well-known L1 scheme is used to solve the subdiffusion equations with weak singularity, numerical simulations show that this scheme can produce  various convergence rates for different choices of model parameters (i.e., domain size, final time $T$, and reaction coefficient $\kappa$). In fact, this elusive phenomenon can be found  in other numerical methods for reaction-diffusion equations such as the backward Euler (IE) scheme, Crank-Nicolson (C-N) scheme, and BDF2 scheme. The current theory in the literatures cannot explain why there exists two different convergence regimes, which has been puzzling us for a long while, and motivating us to study  this inconsistence between the  standard convergence theory and numerical experiences.  In this talk, we provide a general methodology  to systematically obtain error estimates that incorporate the exponential decaying feature of the solution. We call this novel error estimate decay-preserving error estimate and apply it to aforementioned IE, C-N, and BDF2 schemes. Our estimates  reveal  that  the various convergence rates are caused by the  trade-off between the two  components in different model parameter regimes . In this way, we are able to capture different states of  the convergence rate, for which the traditional error estimates fail, since we take the model parameters into account and thus retain more properties of  the continuous solution. In addition, the alpha-robust estimates for L1 and Alikhanov's schemes on general nonuniform meshes are also reported. The works are jointed with Zhimin Zhang and Chengchao Zhao. 

张继伟，武汉大学数学与统计学院教授，博士生导师。2009年在香港浸会大学获得博士学位，随后在南洋理工大学和纽约大学克朗所从事博士后研究，2014年5月在北京计算科学研究中心工作，2018年11月到武汉大学工作。主要研究领域包括偏微分方程和非局部模型的数值解法，以及神经科学的建模与计算，得到了国家自然科学基金重点项目、面上项目以及基础加强向等项目的支持。在包括SINUM，SISC，MCOM，JCNS等国际知名期刊上发表学术论文80余篇。