Numerische Mathematik

, Volume 50, Issue 1, pp 1–15

A finite element method for a singularly perturbed boundary value problem

  • Martin Stynes
  • Eugene O'Riordan
Article

DOI: 10.1007/BF01389664

Cite this article as:
Stynes, M. & O'Riordan, E. Numer. Math. (1986) 50: 1. doi:10.1007/BF01389664

Summary

We examine the problem:εu″+a(x)u′−b(x)u=f(x) for 0<x<1,a(x)≧α>0,b(x)>β,α2 = 4εβ>0,a, b andf inC2 [0, 1], ε in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh2, whereC is independent ofh and ε). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (“hat” functions), uniform first order accuracy is obtained in theL1 (0, 1) norm.

Subject Classifications

AMS(MOS): 65L10CR: G.1.7

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Martin Stynes
    • 1
  • Eugene O'Riordan
    • 2
  1. 1.Department of MathematicsUniversity CollegeCorkIreland
  2. 2.Department of MathematicsDundalk Regional Technical CollegeDundalkIreland