Abstract. We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations should allow one, in particular, to employ as trial spaces directional representation systems. Since such systems are known to be stable in L 2 special emphasis is placed on L 2 -stable formulations. The proposed stability concept is based on perturbations of certain "ideal" test spaces in Petrov-Galerkin formulations. We develop a general strategy for realizing corresponding schemes without actually computing excessively expensive test basis functions. Moreover, we develop adaptive solution concepts with provable error reduction. The results are illustrated by first numerical experiments.

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

Abstract. In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli's work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.Mathematics Subject Classification. 65N12, 35J50, 65N30.

No abstract

This paper considers the Dirichlet problem, where a is a scalar diffusion function. For a fixed f , we discuss under which conditions is a uniquely determined and when can a be stably recovered from the knowledge of u a . A first result is that whenever a ∈ H 1 (D), with 0 < λ ≤ a ≤ Λ on D, and f ∈ L ∞ (D) is strictly positive, thenMore generally, it is shown that the assumption a ∈ H 1 (D) can be weakened to a ∈ H s (D), for certain s < 1, at the expense of lowering the exponent 1/6 to a value that depends on s.