TY - JOUR T1 - Numerical Methods for the Extended Fisher-Kolmogorov (EFK) Equation JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 186 EP - 210 PY - 2006 DA - 2006/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/896.html KW - 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. AB -

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.