We consider the numerical solution, by a Petrov–Galerkin finite-element method, of a singularly perturbed reaction–diffusion differential equation posed on the unit square. In Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 50, 2729–2743), it is argued that the natural energy norm, associated with a standard Galerk...

In this paper a two-point boundary value problem for a system of singularly perturbed ordinary differential equations is considered. It is shown that a parameter robust method can be constructed which gives first order convergence in the maximum norm.

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu′′+2u′+f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slo...

In these notes we describe some methods for studying the asymptotic behavior of solutions to a class of singularly perturbed elliptic problems. We present first the case of concentration at single points, and then at sets of positive dimension.

In this paper we construct the asymptotics of solution Cauchy problem for a singularly perturbed hyperbolic system by using regularization method problems S.A. Lomov. The Lomov is used first time to asymptotic system.

Spline methods provide an important tool to solve singularly perturbed boundary value problems. The present paper gives a comprehensive review of computational methods based on splines used in the solution of various classes of singularly perturbed problems such as self adjoint, linear, non-linear, semi-linear, quasi-linear, singular, single parameter and multi parameter. The spline based numer...