A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory

Abstract

In this paper, we discuss a class of nonlocal parabolic systems with nonlinear boundary conditions arising from the thermal explosion theory. First, we prove the local existence and uniqueness of the classical solution using the Leray–Schauder fixed-point theorem. Then, we analyze three Galerkin approximations of the system and derive the optimal-order error estimates: $\mathcal{O}(h^{r+1})$ in $L^2$ norm for continuous-time Galerkin approximation, $\mathcal{O}(h^{r+1}+{(\Delta t)}^2)$ in the $L^2$ norm for Crank–Nicolson Galerkin approximation, and $\mathcal{O}(h^{r+1}+{(\Delta t)}^2)$ in both $L^2$ and $H^1$ norms for extrapolated Crank–Nicolson Galerkin approximation.