A parameter‐robust numerical method for a system of reaction–diffusion equations in two dimensions

Abstract

A system of M(≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite-difference method for the solution of the system on a Shishkin mesh is presented, and it is proved that the computed solution is second-order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data.

A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions

“…[5] and [4,Section 3]. Nevertheless, unlike [4], the operator L does not obey a maximum principle, and consequently our analysis is very different from that of [4].…”

“…The analysis of singularly perturbed reaction-diffusion problems on Shishkin meshes on the domain Ω has been carried out by Clavero et al [2] in the case of a single equation and by Kellogg et al [4] for a system of equations, but the error analysis in both papers relies on the lengthy construction of a decomposition of the solution. The analysis for Shishkin meshes in the present paper is much simpler.…”

“…A problem resembling (1.1) is analysed in [4], but the hypotheses there ensure that A is diagonally dominant and inverse-monotone so that the system satisfies a maximum principle, while in the current paper the only hypothesis on A is Assumption 1.1 below and we make no appeal to maximum principles for systems. Thus [4] and the current paper consider distinct but overlapping classes of problems.…”

“…Thus [4] and the current paper consider distinct but overlapping classes of problems. While the numerical method of [4] is identical with that of the current paper, the analysis of [4] relies heavily on the maximum principle already mentioned, and on the construction of an elaborate decomposition of u into a sum of a smooth part, a layer component associated with each edge ofΩ, and a corner layer associated with each corner ofΩ. A similar decomposition is used even in the case of a single reaction-diffusion equation in [2].…”

A coupled system of singularly perturbed parabolic reaction-diffusion equations

“…In the literature there are numerous papers dealing with the steady-state [1,7,8,[10][11][12][13][14][15]21] version of problem (1.1). Bakhvalov [1] established second order convergence under the assumptions that ε i = ε, ∀i and that the coupling matrix A(x) was coercive.…”