Finite Elements and Practical Error Analysis of Huxley and EFK Equations

In this dissertation, long time error estimates are obtained using non-traditional methods for the Hodgkin-Huxley equation

u_t + u_xx = u(1-u)(u-a) for a ∈ (0,1/2)

u_t + γ u _xxxx - u_xx = u-u³

Traditional methods for analyzing exact error propagation depends on the stability of the numerical method employed. Whereas, in this dissertation the analysis of the exact error propagation uses evolving attractors and only depends on the stability of the dynamical system. The use of the smoothing indicator yields a posteriori estimates on the numerical error instead of a priori estimates.