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We consider the stationary reaction-diffusion model in a domain Ω ∈ R n having the size along one coordinate direction essentially smaller than along the others. By an energy type argumentation we deduce two simplified models of lower dimension (zero-, and first-order models), which are defined on a domain of the dimension n − 1. For these models, we derive fully computable estimates of the difference between the solution of the original problem and n−dimensional reconstructions generated by solutions of dimensionally reduced problems.

-A functional type a posteriori error estimator for the finite element discretization of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimization problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.2010 Mathematical subject classification: 65N30, 65N15, 65J15, 65N22, 65J10.

We consider Poisson's equation in an n-dimensional exterior domain G(n≥2) with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certain Lq(G)-spaces there exists a solution in the homogeneous Sobolev space S2,q(G), containing functions being local in Lq(G) and having second-order derivatives in Lq(G) Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimension n+1, independent of q

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