Efficient approximation of the solution of certain nonlinear reaction–diffusion equations with small absorption

Abstract

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the absorption is small enough, compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the "continuous" equation. Furthermore, we exhibit an algorithm computing an ε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.

Newton’s method and a mesh-independence principle for certain semilinear boundary-value problems

“…It is clear that Theorem 2 cannot be applied to show the convergence of the Newton sequence (9), because the hypothesis of convexity of the domain D of the statement of Theorem 2 is not satisfied by the set D ⊂ X defined above. We shall nevertheless obtain a variant of Theorem 2 which proves that, starting at an arbitrary element z ∈ D, the Newton sequence (9) converges.…”

“…For a given mesh size, the solutions of the discretization of (1) were studied in [7,3,4,8,9], for different conditions concerning g and f . In this paper we consider the behavior of the discretization of the solutions (1) as the mesh size tends to zero, that is, we aim to approximate the solutions of (1).…”