We present a simple and easy-to-use Nash–Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter ε → 0. The novel feature is to allow loss of powers of ε as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the util...

In this work, superconvergent approximation of singularly perturbed two-point boundary value problems of reaction-diiusion type and convection-diiusion type are studied. By applying the standard nite element method on the Shishkin mesh, superconvergent error bounds of (N ?1 ln(N +1)) p+1 in a discrete energy norm are established. The error bounds are uniformly valid with respect to the singular...

Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of gradient type, in which a small parameter conveys a singular perturbation. The physical motivation comes from a multi-order-parameter phase field model developed by Braun et al [BCMcFW] and [T] for the description of crystalline interphase boundaries. In thi...

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, and the renormalization group method are well known. It is shown that all of the solutions of the reduced equatio...

On the conditioning of an upwind nite-diierence approximation of a convection-diiusion boundary value problem on an adaptive grid. We derive condition number estimates for the standard rst-order upwind numerical discretisation of a model convection-diiusion boundary value problem on a non-uniform grid. The grid is suggested by the equidistribution of a power of the solution gradient. It is show...