Exponential replacement in finite element method for advection-diffusion equations

Abstract

A mathematical model of drug distribution in the artery wall during catheter treatment of
atherosclerosis, which is presented as initial-boundary value problem for a system of two
differential equations, is formulated. During the first numerical experiment it was found that direct
application of the finite element method with standard linear and quadratic basis functions leads
to a loss of stability of the solution. This is due to the specifics of the input parameters of the
problem, in fact a significant advantage over advection coefficients of diffusion coefficients. The
drawback is overcome by using approximations based on exponential replacement in problem
formulation that leads to a loss of advection term and after by using reverse replacement inside
finite element method. Results of computational experiments for one-dimensional spatial variables
for stationary problems are demonstrated.