Numerical approximation of the two-species Shigesada-Kawasaki-Teramoto model

Abstract

In this article, the finite element approximation of the two-species Shigesada-Kawasaki-Teramoto model, with cross-diffusion for one species, is studied. In the beginning, the regularized problem of the model and the entropy proof of the regularized problem were presented. Also, an approxi- mation of the regularized problem using the finite element method was introduced. A number of bounds were found in different spaces for approximate solutions. In addition, depending on the bounds that are obtained, the convergence of the approximate problem to the weak form of the problem was studied, and thus the existence of solutions to the Shigesada-Kawasaki-Teramoto model was proved. Also, a practical algorithm for solving the approximate problem was presented and used to calculate the numerical results of the problem.