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

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, SIAM J. Numer. Anal., 39, 1446-1467) 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 ε². Numerical experiments are presented that support our theoretical results.