## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

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#### Year of publication

- 2000 (16) (remove)

#### Document Type

- Preprint (16) (remove)

#### Keywords

- CFL type conditions (1)
- Fokker-Planck equation (1)
- Grad expansion (1)
- Gravimetry (1)
- Ill-posed problem (1)
- Integral Equations (1)
- Inverse Problem (1)
- Isotropy (1)
- Multiscale Methods (1)
- Numerics (1)

- 234
- A spectrum preserving collocation scheme for Slender-Body Approximations to Stokes Flow (2000)
- Linearized flows past slender bodies can be asymptotically described by a linear Fredholm integral equation. A collocation method to solve this equation is presented. In cases where the spectral representation of the integral operator is explicitly known, the collocation method recovers the spectrum of the continuous operator. The approximation error is estimated for two discretizations of the integral operator and the convergence is proved. The collocation scheme is validated in several test cases and extended to situations where the spectrum is not explicit.

- 233
- On the Multiscale Solution of Satellite Problems by Use of Locally Supported Kernel Functions Corresponding to Equidistributed Data on Spherical Orbits (2000)
- Being interested in (rotation-)invariant pseudodi erential equations of satellite problems corresponding to spherical orbits, we are reasonably led to generating kernels that depend only on the spherical distance, i. e. in the language of modern constructive approximation form spherical radial basis functions. In this paper approximate identities generated by such (rotation-invariant) kernels which are additionally locally supported are investigated in detail from theoretical as well as numerical point of view. So-called spherical di erence wavelets are introduced. The wavelet transforms are evaluated by the use of a numerical integration rule, that is based on Weyl's law of equidistribution. This approximate formula is constructed such that it can cope with millions of (satellite) data. The approximation error is estimated on the orbital sphere. Finally, we apply the developed theory to the problems of satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG).

- 232
- Basic Aspects of Geopotential Field Approximation From Satellite-to-Satellite Tracking Data (2000)
- The satellite-to-satellite tracking (SST) problems are characterized from mathematical point of view. Uniqueness results are formulated. Moreover, the basic relations are developed between (scalar) approximation of the earth's gravitational potential by "scalar basis systems" and (vectorial) approximation of the gravitational eld by "vectorial basis systems". Finally, the mathematical justication is given for approximating the external geopotential field by finite linear combinations of certain gradient fields (for example, gradient fields of multi-poles) consistent to a given set of SST data.

- 231
- Uniform Stability of a Finite Difference Scheme for Transport Equations in Diffusive Regimes (2000)
- An asymptotic preserving numerical scheme (with respect to diffusion scalings) for a linear transport equation is investigated. The scheme is adopted from a class of recently developped schemes. Stability is proven uniformly in the mean free path under a CFL type condition turning into a parabolic CFL condition in the diffusion limit.

- 230
- On the minimal energy state of a mixture of charged classical and quantum fluids (2000)
- The paper concerns the equilibrium state of ultra small semiconductor devices. Due to the quantum drift diffusion model, electrons and holes behave as a mixture of charged quantum fluids. Typically the involved scaled Plancks constants of holes, \(\xi\), is significantly smaller than the scaled Plancks constant of electrons. By setting formally \(\xi=0\) a well-posed differential-algebraic system arises. Existence and uniqueness of an equilibrium solution is proved. A rigorous asymptotic analysis shows that this equilibrium solution is the limit (in a rather strong sense) of quantum systems as \(\xi \to 0\). In particular the ground state energies of the quantum systems converge to the ground state energy of the differential-algebraic system as \(\xi \to 0\).

- 229
- On the approximation of kinetic equations by moment systems (2000)
- The aim of this article is to show that moment approximations of kinetic equations based on a Maximum Entropy approach can suffer from severe drawbacks if the kinetic velocity space is unbounded. As example, we study the Fokker Planck equation where explicit expressions for the moments of solutions to Riemann problems can be derived. The quality of the closure relation obtained from the Maximum Entropy approach as well as the Hermite/Grad approach is studied in the case of five moments. It turns out that the Maximum Entropy closure is even singular in equilibrium states while the Hermite/Grad closure behaves reasonably. In particular, the admissible moments may lead to arbitrary large speeds of propagation, even for initial data arbitrary close to global eqilibrium.

- 228
- On singular limits of mean-field equations (2000)
- Mean field equations arise as steady state versions of convection-diffusion systems where the convective field is determined as solution of a Poisson equation whose right hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of 2 convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean field equation by a variational technique. Also we analyse the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.

- 227
- Performance of some preconditioners for the p - and hp -version of the finite element method in 3D (2000)

- 226
- Consistency analysis of mesh-free methods for conservation laws (2000)
- Based on general partitions of unity and standard numerical flux functions, a class of mesh-free methods for conservation laws is derived. A Lax-Wendroff type consistency analysis is carried out for the general case of moving partition functions. The analysis leads to a set of conditions which are checked for the finite volume particle method FVPM. As a by-product, classical finite volume schemes are recovered in the approach for special choices of the partition of unity.