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 Jan 25 02:30:58 PM MST 2024