Non-classicalH 1 projection and Galerkin methods for non-linear parabolic integro-differential equations

Abstract

In this paper the Galerkin method is analyzed for the following nonlinear integro-differential equation of parabolic type: $$c(u)u_t = \nabla \cdot \{ a(u)\nabla u + \int_0^t {b(x, t, r, u(x, r))} \nabla u(x, r) dr\} + f (u)$$ Optimal L 2 error estimates for Crank-Nicolson and extrapolated Crank-Nicolson approximations are derived by using a non-classicalH 1 projection associated with the above equation. Both schemes result in procedures which are second order correct in time, but the latter requires the solution of a linear algebraic system only once per time step.