报告学者：Natalia Kopteva

报告者单位：University of Limerick

报告时间：7月3日16:00

报告地点：SX106

报告摘要:Over the past decade, there has been a growing interest in evolution equations of parabolic type that involve fractional-order derivatives in time of order in (0, 1). Such equations, also called subdiffusion equations, arise in various applications in engineering, physics, biology and finance. Hence, it is quite important to develop efficient and reliable computational tools for their numerical solution. 

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, building on some ideas from Kopteva Math. Comp. 19, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. This approach is employed in the error analysis of the L1 and Alikhanov L2-1${}_\sigma$  fractional-derivative operators, as well as an L2-type discretization of order $3-\alpha$ in time. This methodology is also generalized for semilinear fractional parabolic equations. In particular, our error bounds accurately predict that milder (compared to the optimal) grading yields optimal convergence rates in positive time. The theoretical findings are illustrated by numerical experiments.

Furthermore, pointwise-in-time a posteriori error bounds will be given in the spatial $L_2$ and $L_\infty$ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates $2-\alpha$ in the presence of solution singularities.

报告学者简介：Natalia Kopteva，在俄罗斯莫斯科大学获得博士学位，曾在俄罗斯莫斯科大学，英国斯特拉斯克莱德大学等任教，现为爱尔兰利莫瑞克大学数学与统计学院教授。Natalia Kopteva教授的主要研究领域是分数阶微分方程及奇异摄动微分方程的数值解法，包括后验误差估计以及各向异性的边界层自适应网格等。目前为SIAM Journal on Numerical Analysis等期刊编委。