Representative problems: Analysis and (High-Fidelity) approximation

Abstract

Partial differential equations (PDEs) represent the foundation upon which many mathematical models for real-life applications are erected. In order to solve these equations one almost invariably has to resort to efficient approximation techniques (such as the finite element method, for example). These are also called high-fidelity approximations, and represent the building blocks of any kind of reduced-order model, such as the reduced basis (RB) method for parametrized PDEs presented in this book. We review the formulation, analysis and approximation of three important classes of variational problems, namely strongly coercive, weakly coercive (also called noncoercive) and saddle-point (also called mixed variational) problems. Each case is accompanied by some examples of interest.