Numerical Methods for the Extended Fisher-Kolmogorov (EFK) Equation

Abstract

In the study of pattern formation in bi-stable systems, the extended Fisher-Kolmogorov (EFK) equation plays an important role. In this paper, some a priori bounds are proved using Lyapunov functional. Further, existence, uniqueness and regularity results for the weak solutions are derived. Using $C^1$-conforming finite element method, optimal error estimates are established for the semidiscrete case. Finally, fully discrete schemes like backward Euler, two step backward difference and Crank-Nicolson methods are proposed, related optimal error estimates are derived and some computational experiments are discussed.