Galerkin Methods for Nonlinear Parabolic Integrodifferential Equations with Nonlinear Boundary Conditions

Abstract

Galerkin methods are analyzed for the nonlinear parabolic integrodiflerential equation \[ u_t = \nabla \cdot \left\{ {a(u)\nabla u + \int_0^t {b(t,u(x,s))\nabla u(x,s)ds} } \right\} + f(u)\] in $\Omega \times ( {0,T} ]$, $T > 0$, $\Omega \subset R^d (d \leqq 3)$, subject to the nonlinear boundary condition \[ a(u)\frac{{\partial u}}{{\partial v }} + \int_0^t {b(t,u(x,s))} \frac{{\partial u}}{{\partial v }}(x,s)ds = g(u)\] on $S_T = \partial \Omega \times [0,T]$ and the usual initial condition. Optimal-order error estimates are derived in $L^2 (\Omega )$ norm for the continuous, Crank–Nicolson, and modified Crank–Nicolson Galerkin approximations. As usual, the latter will yield a system of linear algebraic equations to be solved at each level.