Sinchuk Y. Adaptive schemes of finite elements method for singularly perturbed variation convection-diffusion problems

3. The thesis is devoted to investigation of the FEM schemes for singularly perturbed convection-diffusion problems. The finite elements scheme is constructed for solving one-dimensional convection-diffusion problem. Stability and convergence for the scheme are investigated. The exponential one-step recurrent integration scheme for initial boundary value problems is constructed. The finite elements scheme for two-dimensional singularly perturbed convection-diffusion problem is proposed. That is a natural generalization Il'in-Allen-Southwell scheme. The easily implemented algorithm for generation triangulations of two-dimensional regions is developed. A posteriori errors estimators for finite element method are constructed using solutions of one- and two-dimensional local problems. Based on the estimators h-adaptive scheme is designed. The corresponding software is developed. Efficiency and reliability of the proposed methods are illustrated by numerical experiments.