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Stability and accuracy of adapted finite element methods for singularly perturbed problems. (English) Zbl 1146.65059
The authors study the stability and accuracy of the standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) for the following one-dimensional linear singularly perturbed convection-diffusion two-point boundary-value problems
\[ -\varepsilon u'' - b u' = f, \quad (0,1), \quad u(0) = g_0, \quad u(1) = g_1. \]
It has been shown that the accuracy of the standard FEM depends crucially on the uniformity of the grid away from the boundary layer. Here, the authors develop a new SDFEM based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property. Optimal convergence results for the standard FEM and the new SDFEM are obtained.

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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