TY - JOUR

T1 - A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem

AU - Chadha, Naresh M.

AU - Kopteva, Natalia

PY - 2011/1

Y1 - 2011/1

N2 - 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, SIAM J. Numer. Anal., 39, 1446-1467) 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(|ln ε|/ ln N) 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.

AB - 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, SIAM J. Numer. Anal., 39, 1446-1467) 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(|ln ε|/ ln N) 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.

KW - adaptive mesh

KW - finite differences

KW - grid equidistribution

KW - maximum norm

KW - reaction-diffusion

KW - singular perturbation

UR - http://www.scopus.com/inward/record.url?scp=79251499674&partnerID=8YFLogxK

U2 - 10.1093/imanum/drp033

DO - 10.1093/imanum/drp033

M3 - Article

AN - SCOPUS:79251499674

SN - 0272-4979

VL - 31

SP - 188

EP - 211

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 1

ER -