Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems Eugene O’ Riordan ∗ and Martin Stynes † February 20, 2007 Abstract A system of two coupled singularly perturbed convection–diffusion ordinary differential equa- tions is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a con- ventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their de- pendence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh are shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm. Short title : Strongly coupled convection-diffusion system AMS Mathematics Subject Classification 2000 : 65L10, 65L12, 65L20, 65L70. Keywords : Singularly perturbed, convection-diffusion, coupled system, piecewise-uniform mesh. 1 Introduction The numerical solution of singularly perturbed ordinary differential equations has attracted much attention in the research literature in the past few decades. Special difference schemes, finite element methods and layer-adapted meshes have all been used and analysed in the numerous papers devoted to convection-diffusion problems; see the references listed in [4, 6, 12]. Nevertheless, amidst all this activity, little attention has been paid to the numerical analysis of singularly perturbed systems of convection-diffusion problems that are coupled through their convective (first-order) terms—there are only a few references such as [1, 2, 7, 8, 9]. In the present paper a system of two singularly perturbed linear second-order ordinary differ- ential equations is examined. Similarly to [1], these equations are strongly coupled through their convective terms. Systems of this type appear in optimal control problems and in certain resistance- capacitor electrical circuits; see [5]. Although a convective coupling appears in [2, 7], it is much weaker since in these papers one of the equations can be solved independently of the other. The coupling in [8] is only through the zero-order terms, which is also much weaker than the coupling that we consider here. The strong coupling of the differential equations has an immediate and far-reaching consequence: as Protter and Weinberger [11, p.192] point out, the system does not have a maximum principle in * School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland; eugene.oriordan@dcu.ie † Department of Mathematics, National University of Ireland, Cork, Ireland; m.stynes@ucc.ie 1