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An $$hp$$ finite element method for convection-diffusion problems in one dimension. (English) Zbl 0952.65061
The authors analyze an $$hp$$ ($$h$$ – mesh width, $$p$$ – degree of piecewise polynomial) finite element method for convection-diffusion problems. Due to a generalization of the classical $$\alpha$$-quadratically upwindet and the Hemker test-functions for piecewise linear trial spaces, they prove that the method is stable independently of the viscosity. It appears that the stability depends only weakly on the spectral order, and for a piecewise polynomial trial functions it is possible to construct upwinded test functions that lead to a uniformly stable method. Employing the regularity results proved, the authors obtain the robust exponential convergence of the method together with the uniform stability. All obtained results are illustrated by numerical examples.

##### MSC:
 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L20 Stability and convergence of numerical 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
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