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A note on the energy norm for a singularly perturbed model problem. (English) Zbl 1239.65055
Summary: We consider a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research.

MSC:
65L50 Mesh generation, refinement, and adaptive 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
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