基于有限元方法的反应扩散种群模型斑图数值模拟研究

In reaction-diffusion population models, self-organized spatial patterns may be generated due to the coupling of nonlinear reaction term and linear diffusion term. Research on pattern formations is of important significances to understand and control plant growth, population migration, epidemic spreading, etc. The development of pattern dynamics in reaction-diffusion systems relies heavily on numerical simulation. At present, finite difference method under normal region is mainly used for numerical simulation. It is difficult to study the spatial-temporal dynamic asymptotic behaviors with large scale, long time and complex region limited by finite difference method. Finite element method, as the mainstream method in the numerical solution of partial differential equations, has several advantages over finite difference method. The project will apply finite element method to the numerical simulation of pattern formations in reaction-diffusion systems. With regard to highly nonlinear reaction term, we will study practical discrete schemes with high precision and give a priori error estimates. With regard to complex region and singularity of solution, we will study a posteriori error estimators to guide the adaptive finite element method, to improve the computational speed and accuracy. With regard to reaction-diffusion system with large time delay, we will study practical and efficient finite element discrete schemes as well as a priori and posteriori estimates. Furthermore, we will study the impact of complex region shape and large time delay on the pattern formations.

在反应扩散系统种群模型中，反应的非线性与扩散的线性行为耦合可使系统自发地产生各种空间斑图，对这些斑图态的研究对于认识和控制植物生长、种群迁移、传染病传播等有重要意义。反应扩散系统斑图动力学的研究进展在很大程度上依赖于数值模拟的推动，当前数值模拟主要在规则区域下使用有限差分方法，受差分法的限制，研究大尺度、长时间、空间区域复杂的时空动力学渐进行为相当困难。有限元作为数值求解偏微分方程的主流方法拥有差分法不具备的优点，本项目拟将有限元方法用于反应扩散系统斑图的数值模拟。针对系统反应项高度非线性的特点，研究实用且精度高的离散格式并给出先验误差估计；针对系统空间区域复杂及解有奇性的特点，给出后验误差估计子以指导自适应有限元计算，提高运算速度和精度；针对有大时滞的反应扩散系统，研究实用高效的有限元离散方法及先验误差估计、后验误差估计，在此基础上，研究复杂区域形状及大时滞对斑图结构的影响。