Finite element solution of the reaction-diffusion equation

Abstract

In this study we present the numerical solution o fboundary value problems for the reaction-diffusion equations in 1-d and 2-d that model phenomena such as kinetics and population dynamics.These differential equations are solved nu- merically using the finite element method (FEM).The FEM was chosen due to several desirable properties it possesses and the many advantages it has over other numerical methods.Some of its advantages include its ability to handle complex geometries very well and that it is built on well established Mathemat- ical theory,and that this method solves a wider class of problems than most numerical methods.The Lax-Milgram lemma will be used to prove the existence and uniqueness of the finite element solutions.These solutions are compared with the exact solutions,whenever they exist,in order to examine the accuracy of this method.The adaptive finite element method will be used as a tool for validating the accuracy of theFEM.The convergence of the FEM will be proven only on the real line.