[NA] Olaf Steinbach: Space-time finite and boundary element methods for the wave equation

26 May 2023 12:30 till 13:15 - Location: 01.230 Elektron | Add to my calendar

In this talk we will review some recent results on space-time finite and
boundary element methods for the wave equation. As a first model problem
we consider the inhomogeneous wave equation with zero Dirichlet boundary
and initial conditions. The related space-time variational formulation
follows the standard approach when applying integration by parts in space
and time simultaneously. Note that a space-time tensor-product based
finite element discretization requires a CFL condition to ensure stability.
While the ansatz and test spaces are both subspaces of functions whose
space-time gradient is square integrable, they differ in zero initial
and terminal conditions to be satisfied. When introducing a modified
Hilbert transformation we end up with a Galerkin variational formulation
which is unconditionally stable. This modified Hilbert transformation is
also an essential tool in the formulation of coercive boundary integral
equations for the wave equation. Finally we also consider distributed
optimal control problems subject to the wave equation, and related
space-time least-squares finite and boundary element methods.

The talk is based on joint work with Marco Zank (Vienna), Richard
Löscher (Graz), Carolina Urzua-Torres (Delft), and Daniel Hoonhout (Delft).