Numerical methods for the extended Fisher-Kolmogorov (EFK) equation
Palla Danumjaya 1, Amiya Kumar Pani 11 Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India
Received by the editors January 1, 2004 and, in revised form, March 22, 2004.
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.
AMS subject classifications: 65L20, 65L60, 65L70
Key words: extended Fisher-Kolmogorov (EFK) equation; Lyapunov functional; weak solution; existence; uniqueness and regularity results; finite element method; semidiscrete method; backward Euler; two step backward difference and Crank-Nicolson schemes; optimal estimates
Email: firstname.lastname@example.org (P. Danumjaya), email@example.com (A. K. Pani)