In many applications the processes under consideration can be described by various models that have different levels of complexity. For example, for flows in gas pipeline networks isothermal or non-isothermal models can be used. Depending on the specific flow regime simple models may or may not be appropriate, and it is often hard to predict which model should be used. Thus, we construct solution algorithms that choose the model adaptively based on modeling error estimators. These are quantities that can be computed from the numerical solution and give information on the difference between solutions of different models. Methodologically, these estimators may be based on dual weighted residuals or on (relative) energy arguments.
Model order reduction is a tool to decrease the computational cost for applications where a parametrized PDE problem needs to be solved multiple times for different parameter values. Therefore it is often studied in the context of optimal control or uncertainty quantification. Snapshot-based model order reduction requires a set of representative samples of the solution, which need to be computed in advance and might be space-adaptive, i.e., computed on different spatial meshes. The solution of the reduced-order model is then represented as a linear combination of these snapshots. The respective coefficients are determined by means of a Galerkin projection, based on a weak form of the governing equations. In this way, the reduced-order model inherits both the spatial structure of typical solutions as well as the underlying physics.
