Turchyn Y. Exponential replacements in the finite element method for singular perturbed problems of advection-diffusion-reaction.

This work is devoted to the development of the new scheme of applying finite element method to the singular-perturbed problems of advection-diffusion-reaction. The basic idea of the approach proposed in the dissertation was to use exponential replacement in the initial boundary value problem. Then, for a modified problem, the variational formulation has been obtained and after applying Green’s formula an inverse exponential replacement has been applied. This process has been lead to a modification of the advection-diffusion-reaction variational problem in comparison with the classical approach. It was suggested to discretize the variational problem by the spatial variables using a linear basis. Theoretical studies have been carried out regarding the existence and uniqueness of the solution of the weak formulation of the problem, as well as the order of convergence of the proposed method. Numerical calculations have shown the high efficiency of the proposed method, and verification by calculating the experimental order of convergence has confirmed the results of theoretical studies. A mathematical model describing the process of drug distribution in the vessel wall in the treatment of atherosclerosis was considered as an applied problem. The exponential replacement in the finite element method has been used to find its approximate solutions and the results of calculations have been analyzed.