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Description:
Variational inequalities appear in a variety of practically relevant problems, e.g., in engineering, medicine, or finance. The theory of stationary coercive variational inequalities is quite well-developed. Much less is known for their noncoercive counterpart appearing, e.g., in instationary variational inequalities. In this thesis, noncoercive variational inequalities are analyzed in a variational formulation with (possibly) different trial and test spaces. This setting emerges, e.g., from a space-time formulation of a time-dependent variational inequality in which time is treated as an additional variable within the variational formulation of the problem. We prove well-posedness for the noncoercive variational inequality. Additionally, we derive error estimates for the inequalities and an equivalent saddle-point problem. Since this saddle-point inequality requires a dual variable and a convex cone, we investigate how corresponding spaces have to be chosen such that the obstacle is respected at any time. With this continuous setting at hand, we present sufficient conditions for well-posedness of corresponding discretizations. The error of the continuous solution compared to the discrete solution is numerically investigated, as this error is neglected in the subsequent model reduction. There, the discrete solution is the so-called truth solution from which the reduced basis is derived in terms of certain linear combinations. Several choices exist to construct reduced spaces, in particular to ensure stability within Petrov- Galerkin or saddle-point frameworks. We examine the selection of different combinations for stable reduced spaces and basis generations. The Kolmogorov N-width is numerically investigated (since we are not aware of analytical results), which is the best achievable error for an approximation in terms of a linear space of fixed dimension N . Then, a comparison of two ways to treat time within the reduced basis method for parabolic problems is performed. The pros and cons of each method are ...