The numerical approximation to positive solution for some reaction-diffusion problems

Abstract

A one-dimensional reaction-diffusion problem, with the reaction term of the form \(u^{p}\), \(p>1\) is considered. For \(p=3\), we deduce an independence on the domain (integral) condition and we use this to approximate  the positive solution of the problem by a completely discrete scheme (finite elements – finite differences).

We show that actually this positive solution and its numerical approximation are respectively the stationary unstable hyperbolic points of the continuous and discrete dynamical systems.

Straightforward direct (based on Newton’s method) and indirect approaches of this positive steady state proved to be unsatisfactory. Thus, we suggest a modified indirect algorithm, which takes into account the integral condition, and prove its convergence.

Authors

Călin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Damian Trif
Babeș-Bolyai University

Keywords

semilinear parabolic problem; independence of domain; weak formulation; Galerkin method; discrete problem; Newton method; unstable hyperbolic solution; indirect algorithm; convergence;

References

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Paper coordinates

C.I. Gheorghiu, D. Trif, The numerical approximation to positive solution for some reaction-diffusion problems, PU.M.A., 11 (2000) no. 2, 243-253.

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