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An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques


Authors: Martin Stynes and Eugene O’Riordan
Journal: Math. Comp. 56 (1991), 663-675
MSC: Primary 65L60; Secondary 34E15
DOI: https://doi.org/10.1090/S0025-5718-1991-1068809-4
MathSciNet review: 1068809
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Abstract: We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and $ {L^2}$ norm error bounds. These bounds are uniform in the perturbation parameter. Our proof uses a variation on the classical Aubin-Nitsche argument, which is novel insofar as the $ {L^2}$ bound is obtained independently of the energy norm bound.


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DOI: https://doi.org/10.1090/S0025-5718-1991-1068809-4
Article copyright: © Copyright 1991 American Mathematical Society