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A continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and L-type methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

Dr. Yubin Yan, born in April 8th, 1965, is an Associate Professor in Mathematics in the Department of Mathematical and Physical Sciences at University of Chester, UK.  He obtained his PhD degree in Mathematics in Chalmers University of Technology in 2003 and was the research associate in University of Manchester (2003-2004) and University of Sheffield (2004-2007).  His research area is numerical analysis for the stochastic and deterministic (partial) differential equation, finite element method, and numerical method for fractional differential equation. He introduced a new framework in 2005 for the error estimates of the finite element method for stochastic parabolic equation which is now regarded as the standard reference in this research area. Up to now, he published more than 70 refereed papers on SIAM J. Numerical Analysis, BIT, IMA J. Numerical Analysis, etc. (50 of them are SCI journals). He supervised 3 PhD theses and 30 MSc dissertations. Currently he is supervising 5 PhD students. He is the regular referee for more than 30 scientific journals. He is now in the Editorial boards of several international research journals including Applied Numerical Mathematics, Frontier of Physics, etc. His citation number is 1224 and h-index is 27 in Google Scholar.