Numerical Solutions of Singularly Perturbed Reaction Diffusion Equation with Sobolev Gradients

Critical points related to the singular perturbed reaction diffusion models are calculated using weighted Sobolev gradient method in finite element setting. Performance of different Sobolev gradients has been discussed for varying diffusion coefficient values. A comparison is shown between the weighted and unweighted Sobolev gradients in two and three dimensions. The superiority of the method is also demonstrated by showing comparison with Newton's method. 1. Introduction Many problems in biology and chemistry can be represented in terms of partial differential equations (PDEs). One of the important models is reaction diffusion problems. Much attention has been devoted for the solutions of these problems. In the literature it is shown that numerical solutions of these problems can be computed provided the diffusion coefficients, reaction excitations, and initial and boundary data are given in a deterministic way. The solution of these PDEs is extremely challenging when they has singularly perturbed behavior. In this paper, we discuss the numerical solutions of where is a small and strictly positive parameter called diffusion coefficient, is some two- or three-dimensional region. The Dirichlet boundary conditions are used to solve the equation. Numerous numerical algorithms are designed to solve these kind of systems [1, 2]. We are also using some numerical techniques to solve these systems based on the Sobolev gradient methods. A weighted Sobolev gradient approach is presented, which provides an iterative method for nonlinear elliptic problems. Weighted Sobolev gradients [3] have been used for the solution of nonlinear singular differential equations. It is shown that [4] significant improvement can be achieved by careful considerations of the weighting. Numerous Sobolev norms can be used as a preconditioner strategies. In Sobolev gradient methods, linear operators are formed to improve the condition number in the steepest descent minimization process. The efficiency of Sobolev gradient methods has been shown in many situations, for example, in physics [4–11], image processing [12, 13], geometric modelling [14], material sciences [15–20], Differential Algebraic Equations, (DAEs) [21] and for the solution of integrodifferential equations [22]. We refer [23] for motivation and background for Sobolev gradients. For some applications and open problems in this area, an interesting article is written by Renka and Neuberger [24]. For the computational comparison an Intel Pentium 1.4？GHZ core i3 machine with 1？GB RAM was used. All the programs were written in