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Robust exponential convergence of the $$hp$$ discontinuous Galerkin FEM for convection-diffusion problems in one space dimension. (English) Zbl 0958.65090
This paper deals with the convergence properties of the $$hp$$ discontinuous Galerkin finite element method (FEM) for steady advection-diffusion equations in one space dimension. The authors establish robust exponential convergence of the $$hp$$ discontinuous Galerkin (DG) FEM without stabilization, and with strong enforcement of the essential boundary conditions, provided the FE-mesh is judiciously chosen. Numerical experiments which confirm this robust exponential convergence of the $$hp$$-DG-FEM are presented.

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