High Accuracy Techniques Based Adaptive Finite Element Method For Two-dimensional Elliptic PDEs

Adaptive finite element method,as one of the numerical methods for solving partial differential equations,is effective for the problems show singularity.The exists adaptive finite element methods,when applied to numerical solving the scientific and engineering problems,usually provide lower quality adaptive mesh,not asymptotic exactness error estimators and excessive adaptive iteration steps.This thesis develops an adaptive finite element method based on high accuracy technology for two-dimensional elliptic partial differential equations.By combining high accuracy finite element technology,including super-convergence analysis,gradient recovery,high quality mesh generation and optimization technology,etc.,with the adaptive mesh adjustment,an efficient adaptive strategy and corresponding adaptive algorithm are proposed and applied to solving various benchmark problems.The second chapter considers the posterior error estimation based on gradient recovery and bisection grid refinement method,designs three adaptive finite element algorithms,and tests the numerical performance of these three algorithms through numerical examples.The third chapter considers the posterior error estimation based on gradient recovery and Centroidal Voronoi Delaunay Tessellation(CVDT)grid generation and optimization technology,designs two adaptive algorithms based on CVDT,and tests the numerical performance of these two algorithms through numerical examples.The numerical results show the superiority of the adaptive finite element method based on high accuracy techniques.