Block Monotone Iterative Method for Semilinear Parabolic Equations with Nonlinear Boundary Conditions

Abstract

Two block monotone iterative schemes, called Jacobi and Gauss–Seidel monotone iterations, are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method. These iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations, including a comparison result between them. It is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Some error estimates and a convergence theorem are given, and various sufficient conditions for the construction of upper and lower solutions are obtained. Numerical results are presented for some physical model problems, including some problems with known continuous solutions and two problems with L-shaped and trapezoidal domains.