报告题目 (Title)：The Positive Definiteness of the Integral-Averaged L1 (IAL1) Fractional Derivative Operator and its Application in H¹-norm Analysis of the IAL1 Method (积分平均L1(IAL1)分数阶导算子的正定性及其在IAL1方法H¹范数分析中的应用内)
报告人 (Speaker)：王元明 教授（华东师范大学）
报告时间 (Time)：2025年6月11日(周三) 8:30-10:30
报告地点 (Place)：校本部GJ303
邀请人(Inviter)：李常品、蔡敏
报告摘要：A new positive definiteness result for the integral-averaged L1 (IAL1) fractional-derivative operator is established. It improves the previous positive definiteness results in the literature and plays an important role in the analysis of H¹-norm error of the IAL1 method. Using this new positive definiteness result, we give an H¹-norm analysis of the stability and convergence of the IAL1 method for a time-fractional diffusion problem with a Caputo time-fractional derivative of order α∈(0,1) on nonuniform time meshes. The H¹-norm stability holds for the general nonuniform time meshes, while the H¹-norm convergence is proved for the time graded meshes and the H¹-norm convergence order in time is min{3 + α,γα}/2 for all α∈(0, 1), where γ ≥ 1 is the mesh grading parameter. Two full discretization methods using finite differences and finite elements in space are considered. The theoretical results are illustrated by numerical results.