GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
A mountain pass method for the numerical solution of semilinear elliptic problems
Nonlinear Analysis: Theory, Methods & Applications
Integrated space-time adaptive hp-refinement methods for parabolic systems
Applied Numerical Mathematics
Journal of Computational Physics
Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems
SIAM Journal on Scientific Computing
NITSOL: A Newton Iterative Solver for Nonlinear Systems
SIAM Journal on Scientific Computing
Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem
Applied Numerical Mathematics
hp FEM for Reaction-Diffusion Equations I: Robust Exponential Convergence
SIAM Journal on Numerical Analysis
A Study of Monitor Functions for Two-Dimensional Adaptive Mesh Generation
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
ACM Transactions on Mathematical Software (TOMS)
Applied Numerical Mathematics
Uniform Pointwise Convergence on Shishkin-Type Meshes for Quasi-Linear Convection-Diffusion Problems
SIAM Journal on Numerical Analysis
Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems
Mathematics of Computation
Applied Numerical Mathematics
Applied Numerical Mathematics
| Hi-index | 31.45 |
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 @e. Two strategies allow adaptive selection of @e. The first depends on the rate of convergence of the nonlinear solver and the second implements backtracking in @e. Finally a simple method is used to select the initial @e. Several examples illustrate the effectiveness of the algorithm.