Solving Regularly and Singularly Perturbed Reaction-Diffusion Equations in Three Space Dimensions

In ``effects of basis selection and h-refinement on error estimator reliability and solution efficiency for high-order methods in three space dimensions'' a fixed high-order h-refinement finite element algorithm, Href was introduced for solving reaction-diffusion equations in three space dimensions.  In this paper Href is coupled with continuation creating an automatic method for solving regularly and singularly perturbed reaction-diffusion equations.  The simple quasilinear Newton solver is replaced by the nonlinear solver NITSOL.  Good initial guesses for the nonlinear solver are obtained by continuation in the small parameter.  Two strategies allow adaptive selection of the small parameter.  The first depends on the rate of convergence of the nonlinear solver and the second implements backtracking in the small parameter.  Finally a simple method is used to select the inital value of the small parameter.  Several examples illustrate the effectiveness of the algorithm.