A numerical study of an adaptive finite element method of lines approach for coupled reaction-diffusion equations in Omega - partialOmega.

A numerical study of an adaptive finite element method of lines (AFEMOL) approach is presented for the approximation of the solution of a system of reaction-diffusion equations coupling species defined on a 2-dimensional domain Ω and species confined to the boundary of the domain ∂Ω. In order to bound the energy norm of the space discretization error, in the AFEMOL the spatial mesh changes automatically at selected times when the underlying triangulation is refined in areas where it is needed. The decision of when and where to modify the mesh is based on the estimation of the space discretization error. The adaptive process and the a-posteriori explicit error estimation exploited in this note are a modification of the pioneer work developed by Bieterman and Babuschka in [Numer. Math. 40 (1982), 339], [Numer. Math. 40 (1982), 373], [J. Comput. Phys. 63 (1986), 33]. The primary interest, in the manuscript, is the effect of the coupling Ω–∂Ω on the performance of the error estimator and the successive adaptive process. Our numerical results indicate that the global error estimators are accurate, the local error indicators are reliable and that the adaptive strategy successfully controls the space discretization error.