Finite element approximation of a reaction-diffusion equation. I. Application of topological techniques to the analysis of asymptotic behavior of the semidiscrete approximations

The initial-boundary value problem for a reaction-diffusion equation \[ {u_t} = {u_{xx}} + f\left ( u \right ), \qquad f\left ( u \right ) = - u\left ( {u - b} \right )\left ( {u - 1} \right ), \qquad 0 < b < 1/2, \qquad \left ( * \right )\] was analyzed in [4, 17] by using the Conley index. In this paper we study the asymptotic behavior of the semidiscrete finite element approximations, with interpolation of the coefficients in the nonlinear terms. We show that for small $h$ the spectrum of the linearized discrete steady-state problem is a “good” approximation for the spectrum of the linearized continuous steady-state problem. Using the interpretation of the Conley index as the dimension of an unstable manifold of a steady-state solution, we establish that the properties of the semidiscrete approximations are completely analogous to those of the solutions of $\left ( * \right )$. The asymptotic, as $t \to \infty$, optimal order convergence of the approximate solutions is proved.