Non-classical \(H^ 1\) projection and Galerkin methods for non-linear parabolic integro-differential equations (Q1262108)

The initial-boundary value problem for the equation \[ c(u)u_ t=\nabla \cdot \{a(u)\nabla u+\int^{t}_{0}b(x,t,r,u(x,r))\nabla u(x,r)dr\}+f(u\quad) \] is treated by Crank-Nicolson and extrapolated Crank-Nicolson approximations by using a non-classical \(H^ 1\) projection method. Optimal \(L^ 2\) error estimates are derived, and the schemes are shown to have second order accuracy in time. An advantage of the second scheme is that at each time-step only a linear algebraic system of equations is to be solved.