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

Dr. Marco Zank

Nov. 21, 2019, 1:45 p.m. S2 416-1

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).