###
Wolfgang Bangerth

** Adaptive Finite-Elemente-Methoden zur
Lösung der Wellengleichung mit Anwendung in der
Physik der Sonne**

(English title: "Adaptive finite element methods for the
solution of the wave equation with application to solar
physics")

Thesis, University of Heidelberg, 1998 (in
German).

In this work, adaptive concepts for the numerical solution of the wave
equation in inhomogeneous media are derived and applied to an example
taken from the physics of the solar atmosphere. The main focus is on ways
to estimate the error in the numerical solution with regard to arbitrary
functionals, i.e. quantities of interest, and the use of these estimates
for the generation of computational meshes best suited for the evaluation
of this functional.

Advantages and difficulties of this method are presented. In particular,
it is shown that the proposed approach is significantly better in many
cases than previous adaptive schemes not taking into account the quantity
of interest.Cases involving nonlinear functionals and in which the
approach fails, are presented along with theoretical explanations and
numerical evidence of the reasons for this.

The proposed methods are applied to a simple model from the physics of the
solar atmosphere and the propagation of linear acoustic waves is computed.
The fraction of the wave energy that passes the chromosphere-corona
transition is computed to good accuracy.

*Wolfgang Bangerth*

* Thu May 4 14:55:17 MDT 2017 *