Solving regularly and singularly perturbed reaction-diffusion equations in three space dimensions

In [P.K. Moore, Effects of basis selection and h-refinement on error estimator reliability and solution efficiency for higher-order methods in three space dimensions, Int. J. Numer. Anal. Mod. 3 (2006) 21–51] 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 of Moore, (2006) is replaced by the nonlinear solver NITSOL [M. Pernice, H.F. Walker, NITSOL: a Newton iterative solver for nonlinear systems, SIAM J. Sci. Comput. 19 (1998) 302–318]. Good initial guesses for the nonlinear solver are obtained using continuation in the small parameter ϵ. Two strategies allow adaptive selection of ϵ. The first depends on the rate of convergence of the nonlinear solver and the second implements backtracking in ϵ. Finally a simple method is used to select the initial ϵ. Several examples illustrate the effectiveness of the algorithm.