Space-Time Methods: Modeling, Discretization, Numerical Solution, Error-Controlled Adaptivity, and Applications

The objective of this class is to give an introduction to the mathematical-numerical treatment of space-time formulations of differential equations. Space-time methods treat space and time together in a space-time domain (often called space-time cylinder), which allow us to treat the full problem with a common functional framework (solution sets, function spaces, and weak formulations), same basic discretization (e.g., Galerkin finite elements), joint numerical nonlinear and linear solution, common concepts for a priori and a posteriori error estimation, up to consistent adjoint derivations for gradient-based optimization and goal-oriented error control with the dual-weighted residual method. The focus is on numerical concepts, which are extensively derived with great care and from which most are rigorously proven. Many parts are complemented with numerical examples, figures and loving illustrations.