Singularly perturbed reaction–diffusion equations in a circle with numerical applications

Abstract

The goal of this article is to study the boundary layers of reaction–diffusion equations in a circle and provide some numerical applications which utilize the so-called boundary layer elements. Via the boundary layer analysis, we obtain the valid asymptotic expansions at any order and devise boundary layer elements to be conveniently used in the finite element schemes. Using boundary layer elements incorporated in the finite element space, we obtain accurate numerical solutions in a quasi-uniform mesh with convergence of order 2.

Keywords:

• singular perturbations
• asymptotic analysis
• boundary layer
• finite element method
• reaction–diffusion equations

Keywords:

• 34E15
• 80M35
• 65D10
• 65N12
• 65N30

The authors would like to thank Professor Roger Temam for his suggestions of the problems and constant support during this research. This work was supported in part by NSF Grants DMS 0906440 and DMS 1206438 and by the Research Fund of Indiana University and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012001167).