An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques. (English) Zbl 0718.65062

Stynes, Martin; O’Riordan, Eugene

Math. Comput. 56, No. 194, 663-675 (1991).

This paper gives 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, energy norm and \(L^ 2\) norm error bounds are derived. These bounds are uniform in the perturbation parameter. The 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.

Reviewer: I.N.Katz (St.Louis)

MSC:

65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10	Numerical solution of boundary value problems involving ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34E15	Singular perturbations for ordinary differential equations

Keywords:

singular perturbation; Aubin-Nitsche lemma; Petrov-Galerkin finite element methods; error bounds