Finite element analysis of extended Fisher-Kolmogorov equation with Neumann boundary conditions

Abstract

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R⊂Rd, where d≤3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings.