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) 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(| lnε|/ lnN) 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.