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.