For the discretisation of time-dependent partial differential equations, the classical approaches are time stepping schemes together with finite element methods in space. An alternative is to discretise the time-dependent problem without separating the temporal and spatial variables. However, space-time approximation methods depend strongly on the space-time variational formulations on the continuous level. The focus of this work is on space-time variational formulations for the heat and wave equation, which result not only in inf-sup stable formulations but fit also very well to conforming space-time discretisations.
The first part investigates the heat equation in anisotropic Sobolev spaces, where a type of Hilbert transform is introduced such that ansatz and test spaces are equal. Unconditional stability is proven for any conforming discretisation of this space-time variational formulation.
The second part considers space-time variational formulations for the wave equation. New existence and uniqueness results for the wave equation in a weak and in a strong sense are proven, including isomorphic solution operators and corresponding inf-sup conditions. In addition, an unconditionally stable space-time finite element method with piecewise linear, continuous functions is derived.