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Wolfgang Bangerth, Marcus
Grote and Christel Hohenegger

**Finite Element Method For Time Dependent
Scattering: Nonreflecting
Boundary Condition, Adaptivity, and Energy Decay**

Computer Methods in Applied Mechanics and Engineering,
vol. 193 (2004), pp. 2453-2482.

An adaptive finite element method is developed for acoustic wave propagation
in unbounded media. The efficiency and high accuracy of the method are
achieved by combining an exact nonreflecting boundary condition
[GK95,GK96] with space-time adaptivity [BR99b]. Hence the
computational effort is concentrated where needed, while the artificial
boundary can be brought as close as desired to the scatterer. Both features
combined yield high accuracy and keep the number of unknowns to a minimum.
An energy inequality is derived for the initial-boundary value problem at
the continuous level. Together with an implicit second order time
discretization it guarantees unconditional stability of the
semi-discrete system. The resulting fully discrete linear system
that needs to be solved every time step is unsymmetric but can
be transformed into an equivalent sequence of small nonsymmetric
and large symmetric positive definite systems, which are
efficiently solved by conjugate gradient methods.
Numerical examples illustrate the high accuracy of the
method, in particular in the presence of complex geometry.

*Wolfgang Bangerth*

* Sun Aug 20 15:19:52 MDT 2017 *