Numerical Solutions of Singularly Perturbed Reaction Diffusion Equation with Sobolev Gradients

Abstract

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.

The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

“…The standard implicit Euler method and the HODIE compact fourth order finite difference scheme were used [14]. Sobolev gradient methods were used with Dirichlet boundary conditions [15]. By taking into account the finite element method, sign-changing solutions are obtained [16].…”