## Fachbereich Mathematik

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- Galerkin-Methode (4) (remove)

We analyze the regular oblique boundary problem for the Poisson equation on a C^1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax-Milgram lemma enables us to prove an existence and uniqueness result for weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples we show the applicability of our results. Finally a Ritz-Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.

The following three papers present recent developments in nonlinear Galerkin schemes for solving the spherical Navier-Stokes equation, in wavelet theory based on the 3-dimensional ball, and in multiscale solutions of the Poisson equation inside the ball, that have been presented at the 76th GAMM Annual Meeting in Luxemburg. Part A: A Nonlinear Galerkin Scheme Involving Vectorial and Tensorial Spherical Wavelets for Solving the Incompressible Navier-Stokes Equation on the Sphere The spherical Navier-Stokes equation plays a fundamental role in meteorology by modelling meso-scale (stratified) atmospherical flows. This article introduces a wavelet based nonlinear Galerkin method applied to the Navier-Stokes equation on the rotating sphere. In detail, this scheme is implemented by using divergence free vectorial spherical wavelets, and its convergence is proven. To improve numerical efficiency an extension of the spherical panel clustering algorithm to vectorial and tensorial kernels is constructed. This method enables the rapid computation of the wavelet coefficients of the nonlinear advection term. Thereby, we also indicate error estimates. Finally, extensive numerical simulations for the nonlinear interaction of three vortices are presented. Part B: Methods of Resolution for the Poisson Equation on the 3D Ball Within the article at hand, we investigate the Poisson equation solved by an integral operator, originating from an ansatz by Greens functions. This connection between mass distributions and the gravitational force is essential to investigate, especially inside the Earth, where structures and phenomena are not sufficiently known and plumbable. Since the operator stated above does not solve the equation for all square-integrable functions, the solution space will be decomposed by a multiscale analysis in terms of scaling functions. Classical Euclidean wavelet theory appears not to be the appropriate choice. Ansatz functions are chosen to be reflecting the rotational invariance of the ball. In these terms, the operator itself is finally decomposed and replaced by versions more manageable, revealing structural information about itself. Part C: Wavelets on the 3–dimensional Ball In this article wavelets on a ball in R^3 are introduced. Corresponding properties like an approximate identity and decomposition/reconstruction (scale step property) are proved. The advantage of this approach compared to a classical Fourier analysis in orthogonal polynomials is a better localization of the used ansatz functions.

This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier-Stokes equation on the sphere. It extends the work of Debussche, Marion,Shen, Temam et al. from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on type 3 vector spherical harmonics is introduced and convergence is indicated. Further it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-3j coefficients. To improve the numerical efficiency and economy we introduce an FFT based pseudo spectral algorithm for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales with O(N^3), if N denotes the maximal spherical harmonic degree. The latter is demonstrated in an extensive numerical example.

Matrix Compression Methods for the Numerical Solution of Radiative Transfer in Scattering Media
(2002)

Radiative transfer in scattering media is usually described by the radiative transfer equation, an integro-differential equation which describes the propagation of the radiative intensity along a ray. The high dimensionality of the equation leads to a very large number of unknowns when discretizing the equation. This is the major difficulty in its numerical solution. In case of isotropic scattering and diffuse boundaries, the radiative transfer equation can be reformulated into a system of integral equations of the second kind, where the position is the only independent variable. By employing the so-called momentum equation, we derive an integral equation, which is also valid in case of linear anisotropic scattering. This equation is very similar to the equation for the isotropic case: no additional unknowns are introduced and the integral operators involved have very similar mapping properties. The discretization of an integral operator leads to a full matrix. Therefore, due to the large dimension of the matrix in practical applcation, it is not feasible to assemble and store the entire matrix. The so-called matrix compression methods circumvent the assembly of the matrix. Instead, the matrix-vector multiplications needed by iterative solvers are performed only approximately, thus, reducing, the computational complexity tremendously. The kernels of the integral equation describing the radiative transfer are very similar to the kernels of the integral equations occuring in the boundary element method. Therefore, with only slight modifications, the matrix compression methods, developed for the latter are readily applicable to the former. As apposed to the boundary element method, the integral kernels for radiative transfer in absorbing and scattering media involve an exponential decay term. We examine how this decay influences the efficiency of the matrix compression methods. Further, a comparison with the discrete ordinate method shows that discretizing the integral equation may lead to reductions in CPU time and to an improved accuracy especially in case of small absorption and scattering coefficients or if local sources are present.