A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem

Abstract

The numerical solution of a singularly perturbed semilinear reaction-diffusion two-point boundary value problem is addressed. The method considered is adaptive movement of a fixed number (N + 1) of mesh points by equidistribution of a monitor function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001) to a new equation and a more intricate monitor function. It is proved that there exists a solution to the fully discrete equidistribution problem, i.e. a mesh exists that equidistributes the discrete monitor function computed from the discrete solution on this mesh. Furthermore, in the case when the boundary value problem is linear, it is shown that after O(| ln ε|/ ln N) iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves second-order accuracy in the maximum norm, uniformly in the diffusion coefficient ε 2. Numerical experiments are presented that support our theoretical results.

Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction–diffusion boundary-value problems

“…For scalar reaction-diffusion problems, a new monitor function is proposed in [11], which is very similar to the one, obtained by Beckett and Mackenzie [1]. It is seen that the monitor function used in [1] works well for both convection-diffusion as well as reaction-diffusion problems of scalar form (see Chadha and Kopteva [3]) and also for reaction-diffusion problems with Robin type boundary conditions, by Das and Natesan [6]. In this article, our aim is to provide an a priori and a posteriori error estimates using mesh equidistribution technique for a system of reaction-diffusion problems with diffusion parameters of different magnitudes.…”

“…This algorithm is successfully applied by Kopteva et al [12] and Das and Natesan [6] for scalar reaction-diffusion problems and in [8] for convection-diffusion problems. The convergence of this algorithm for convection-diffusion problems can be seen in [13] and for reaction-diffusion problems in [3] (see also [30]). We shall use this algorithm to generate layer adapted meshes by extending it to a system of ordinary differential equations.…”

Approximation of singularly perturbed reaction-diffusion problems by quadratic C 1-splines

“…It is based on an idea by de Boor [4] and uses an equidistribution principle. Its convergence in connection with an error estimator for a central difference scheme was recently studied by Kopteva and Chadha [2].…”

A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems

“…It is based on an idea by de Boor [3] and uses an equidistribution principle. Its convergence in connection with an error estimator for the second order central difference scheme was recently studied by Kopteva and Chadha [2].…”