An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem

An Analysis of a Uniformly Convergent Finite Difference/Finite Element Scheme for a Model Singular-Perturbation Problem*

Uniform tf(h2) convergence is proved for the El-Mistikawy-Werle discretization of the problem —eu" + au' + bu = f on (0,1), u(0) = A, u(l) = B, subject only to the conditions a, b, f e ^r2'oo[0,1] and a(x) > 0, 0 < i < 1. The principal tools used are a certain representation result for the solutions of such problems that is due to the author [Math. Comp., v. 48, 1987, pp. 551-564] and the gener...

Uniformly Convergent Nonconforming Element for 3-d Fourth Order Elliptic Singular Perturbation Problem

where ∆ is the standard Laplace operator, ∂u/∂n denotes the outer normal derivative on ∂Ω and ε is a small real parameter with 0 < ε ≤ 1. This problem can be considered a gross simplification of the stationary Cahn-Hilliard equation with ε being the length of the transition region of phase separation. In particular, we are interested in the regime when ε tends to zero. Obviously, if ε approache...

Institute for Numerical and Computational Analysis Dublin, Ireland An adaptive uniformly convergent numerical method for a semilinear singular perturbation problem

A singularly perturbed semilinear two point boundary value problem is considered, without any restriction on its turning points. A difference scheme is presented for solving this problem on an arbitrary locally quasiuniform mesh. It is shown that the solution of the scheme is first order accurate, uniformly in the perturbation parameter, in a discrete L1 norm. Numerical results are presented fo...

A Global Uniformly Convergent Finite Element Method for a Quasilinear Singularly Perturbed Elliptic Problem

A Globally Uniformly Convergent Finite Element Method for a Singularly Perturbed Elliptic Problem in Two Dimensions

We analyze a new Galerkin finite element method for numerically solving a linear convection-dominated convection-diffusion problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation i ii parameter, of order h ' in a global energy norm which is stronger than the L norm. This order is optimal in this norm for our choice of trial functions.