Do we need decay-preserving error estimate for solving parabolic equations with the initial singularity?
Speaker
Prof. Jiwei Zhang
Wuhan University
Abstract

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.