讲座主题：A Superconvergent Two-Grid Finite Element Method for the Keller-Segel Equations with Cross-Diffusion and Logistic Source

专家姓名：胡宏伶

工作单位：湖南师范大学

讲座时间：2026年11月12日 10:10-11:10

讲座地点：腾讯会议：154-822-176

主办单位：烟台大学数学与信息科学学院

内容摘要：

This paper presents the development and analysis of a fully discrete numerical scheme for a nonlinear parabolic Keller-Segel system, which incorporates additional cross-diffusion and source terms. The model consists of coupled parabolic equations in which the chemotactic flux and the cross-diffusion term in the chemical equation introduce strong nonlinearities and pose significant analytical challenges. Temporal discretization is achieved using a second order backward differentiation formula (BDF2), while spatial discretization is performed with conforming linear triangular finite elements. To effectively handle the nonlinearities, two robust algorithmic strategies are introduced:a Newton iteration method and a novel two-grid approach. The two-grid algorithm is optimized with a fine-to-coarse feedback mechanism to ensure solution state synchronization between grids, effectively mitigating potential cumulative errors in long-term simulations. Compared to the traditional finite element method, the two-grid technique significantly reduces computational cost and CPU time while maintaining accuracy. A rigorous error analysis establishes optimal error estimates for the fully discrete scheme and the two-grid scheme, with the gradient error at mesh nodes achieving a superconvergence rate of O(h2+H4+ τ2). Numerical experiments verify the theoretical convergence orders, confirm the gradient super-convergence, and demonstrate the superior computational efficiency of the proposed algorithms.

主讲人介绍：

胡宏伶，理学博士，湖南师范大学数学与统计学院教授、博士生导师，信息与计算科学系主任，湖南省计算数学应用软件学会常务理事，湖南省数学会理事。主要从事有限元方法及多重网格法研究。主持国家自然科学基金项目4项、湖南省自然科学基金项目2项、教育厅项目2项、长沙市自然科学基金项目1项；入选湖南师范大学“世承人才计划”和“青年优秀人才培养计划”。在Sci China Math、J Comput Math、J Sci Comput、Appl Numer Math等学术期刊发表SCI论文40余篇。