# Parabolic and Hyperbolic PDEs: Space-Time Variational Formulations and Their Discretisations

**Parabolic and Hyperbolic PDEs: Space-Time Variational Formulations and Their Discretisations**

For the discretisation of time-dependent partial differential equations, the standard approaches are explicit or implicit time stepping schemes together with finite element methods in space. An alternative approach is the usage of space-time methods, where the space-time domain is discretised and the resulting global linear system is solved at once. In any case, CFL conditions play a decisive role for stability. In this talk, the model problems are the heat equation and the wave equation.

The first part of the talk investigates the heat equation. The starting point is a space-time variational formulation in anisotropic Sobolev spaces for the heat equation with Dirichlet boundary conditions, where a linear isometry is used such that ansatz and test spaces are equal. A conforming discretisation of this space-time variational formulation in anisotropic Sobolev spaces leads to a Galerkin-Bubnov finite element method, which is unconditionally stable, i.e. no CFL condition is required. However, for the implementation of this method, the realisation of the linear isometry is crucial. Therefore, some comments on possible realisations for piecewise polynomial, globally continuous ansatz and test functions are given. In the end of the first part, numerical examples confirm the theoretical results.

The second part of the talk considers the second-order wave equation. For hyperbolic problems, usually space-time discontinuous Galerkin finite element methods are used, which increase the number of degrees of freedom, and which involve certain parameters to be chosen to ensure stability. Another possibility is the usage of space-time continuous Galerkin finite element methods in connection with a stabilisation. In this talk, the latter is applied as discretisation for the model problem of the scalar second-order wave equation with Dirichlet boundary conditions, which leads to an unconditionally stable method for the tensor-product case. First, a space-time variational formulation of the wave equation and its discretisation via a tensor-product approach including a stabilisation are motivated and discussed. Second, an equivalent variational formulation as a first-order system in space, which can be used for generalisations to unstructured meshes, is given.

In the last part of the talk, numerical examples are shown.

The talk is based on joint work with O. Steinbach (TU Graz).