Adaptive Finite Element Method For Keller-Segel Equations In Two Dimensionals

Keller-segel equations are coupled nonlinear partial differential equations used to describe chemotaxis.In this thesis,we develop an adaptive finite element method for the Keller-Segel equations.Firstly,we consider the a posterior error estimator and adaptive algorithm of the finite element method for linear parabolic equation.And then,we apply the error estimator and adaptive algorithm to the numerical simulation of heat equation and Allen-Cahn equation.The numerical results show the efficiency of the error estimator and the adaptive algorithm.Furthermore,we derive the residual type a posterior error estimator of the finite element method for the Keller-Segel equations,and prove the reliability and efficiency of the error estimator.Finally,the corresponding marking strategy and adaptive finite element algorithm are proposed,and the efficiency of the adaptive finite element algorithm constructed based on the posterior error estimator is verified by numerical examples.