Duality-based adaptivity of model and discretization in multiscale finite-element methods

This thesis develops strategies for a posteriori error control of discretization and model errors, as well as adaptation strategies, in the context of multiscale finite-element methods. This is done within the general methodology of the DualWeighted Residual Method (DWR). In particular, a reformulation of the Heterogeneous Multiscale Method(HMM) as an abstract model-adaptation framework is introduced that explicitly decouples discretization and model parameters. Based on the framework a samplingadaptation strategy is proposed that allows for simultaneous control of discretization and model errors with the help of classical refinement strategies for mesh and sampling regions. Further, a model-adaptation approach is derived that interprets model adaptivity as a minimization problem of a local model-error indicator. This allows for the formulation of an efficient post-processing strategy that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. The proposed framework is tested on an elliptic model problem with heterogeneous coefficients, as well as on an advection-diffusion problem with dominant microscopic transport