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
Mon Jul 17 15:58:29 MDT 2017