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

Abstract

We study the positive stationary solutions of a standard finitedifference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the absorption is large 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.1991 Mathematics Subject Classification. 65H10, 65L10, 65L12, 65H20, 65Y20.

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

“…In [7,3,4,8] we exhibited an algorithm which, for a given j ∈ N and ϵ ′ > 0, computes an ϵ ′ -approximation of the discrete system F j = 0, i.e., a point…”

“…Combining an algorithm of [4] or [8] for the approximation of the discrete solutions of (1) with mesh size h * and estimates provided by our mesh-independence principle we obtain an algorithm which computes a starting point (Theorem 36). Using this starting point and a discrete Newton iteration we obtain an ϵ-approximation of the positive solution of (1) with O((1/ϵ) 1/2 log 2 log 2 (1/ϵ)) flops and function evaluations (Theorem 37).…”

“…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).…”