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

Abstract

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{lunate}. Two strategies allow adaptive selection of ε{lunate}. The first depends on the rate of convergence of the nonlinear solver and the second implements backtracking in ε{lunate}. Finally a simple method is used to select the initial ε{lunate}. Several examples illustrate the effectiveness of the algorithm. © 2006 Elsevier Inc. All rights reserved.