# Efficient algorithms for mathematical models of physical systems (e.g. high-percision geodesy on earth and in space

The aim is to develop (design, analyse and implement) new computational methods for problems in geodesy, meteorology etc. Often geoscience and atmospheric processes are modelled on the sphere using data collected on earth or spacecraft. These problems can be formulated as PDEs or integral equations on a sphere or exterior to a sphere. The inovative computational methods include (i) approximation techniques (Galerkin methods using radial basis functions and spherical splines), (ii) preconditioning techniques based on domain decomposition methods (e.g. overlapping additive Schwarz methods), (iii) adaptive choice of scattered data among millions of data points (e.g. obtained from MAGSAT satellite). When solving elliptic PDEs or corresponding integral equations (hypersingular, weakly singular) on the unit sphere using scattered measured data (with e.g. the Galerkin method) a very ill-conditioned linear system results. For its solution we use additive Schwarz preconditioners; then the so preconditioned system has only a midly growing condition number which does not depend on the scattered data set but rather on the way we decompose the scattered set into smaller sets using radial basis functions.

The Dirichlet-to-Neumann map is used for problems in unbounded domains. A 2D model problem for the Stokes equation is computed in [1] using Taylor Hood finite elements for the mixed formulation. To optimize the algorithm an adaptive scheme based on an a posteriori error estimator in [2] is used (see Fig. 1-2). Further investigations are in progress with Jairo Duque, Universidad del Valle, Cali, Colombia.

Fig.1: Starting mesh                                                     Fig.2: Adaptive refined mesh after 5 iterations

1. M. Andres, Finite Elemente und Dirichlet-zu-Neumann Abbildung mit Anwendungen in der Mechanik, Dipl. Thesis,     2006

2. G.N. Gatica, L.F. Gatica and E.P. Stephan, A FEM-DtN formulation for a non-linear exterior problem in                       incompressible elasticity, Math. Methods Appl. Sci., 26, (2), 2003, pp. 151-170

E. P. Stephan