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This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments. 

Abstract This paper is concerned with a posteriori error bounds for linear transport equations and related questions of contriving corresponding adaptive solution strategies in the context of Discontinuous Petrov Galerkin schemes. After indicating our motivation for this investigation in a wider context the first major part of the paper is devoted to the derivation and analysis of a posteriori error bounds that, under mild conditions on variable convection fields, are efficient and, modulo a data-oscillation term, reliable. In particular, it is shown that these error estimators are computed at a cost that stays uniformly proportional to the problem size. The remaining part of the paper is then concerned with the question whether typical bulk criteria known from adaptive strategies for elliptic problems entail a fixed error reduction rate also in the context of transport equations. This turns out to be significantly more difficult than for elliptic problems and at this point we can give a complete affirmative answer for a single spatial dimension. For the general multidimensional case we provide partial results which we find of interest in their own right. An essential distinction from known concepts is that global arguments enter the issue of error reduction. An important ingredient of the underlying analysis, which is perhaps interesting in its own right, is to relate the derived error indicators to the residuals that naturally arise in related least squares formulations. This reveals a close interrelation between both settings regarding error reduction in the context of adaptive refinements.