Discrete hardy spaces

We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward

Script geometry

In this paper we describe the foundation of a new kind of discrete geometry and calculus called Script Geometry. It allows to work with more general meshes than classic simplicial complexes. We provide the basic denitions as well as several examples, like the Klein bottle and the projective plane. Furthermore, we also introduce the corresponding Dirac and Laplace operators which should lay the groundwork for the development of the corresponding discrete function theory

Hyperbolic Qp-scales

The Qp-scales were first introduced in [1] as interpolation spaces between the Bloch and Dirichlet spaces in the complex space. ... However, such treatment presents the disadvantage of only considering the Euclidean case. In order to obtain an approach to homogeneous hyperbolic manifolds, the projective model of Gel'fand was retaken in [2]. With the help of a convenient fundamental solution for the hyperbolic (homogeneous of degree ®) D® (see [5]) it was introduced in [7] and [3] equivalent Qp scales for homogeneous hyperbolic spaces. In this talk we shall present and study some properties of this hyperbolic scale

Finite element exterior calculus with script geometry

Finite element method is probably the most popular numerical method used in different fields of applications nowadays. While approximation properties of the classical finite element method, as well as its various modifications, are well understood, stability of the method is still a crucial problem in practice. Therefore, alternative approaches based not on an approximation of continuous differential equations, but working directly with discrete structures associated with these equations, have gained an increasing interest in recent years. Finite element exterior calculus is one of such approaches. The finite element exterior calculus utilises tools of algebraic topology, such as de Rham cohomology and Hodge theory, to address the stability of the continuous problem. By its construction, the finite element exterior calculus is limited to triangulation based on simplicial complexes. However, practical applications often require triangulations containing elements of more general shapes. Therefore, it is necessary to extend the finite element exterior calculus to overcome the restriction to simplicial complexes. In this paper, the script geometry, a recently introduced new kind of discrete geometry and calculus, is used as a basis for the further extension of the finite element exterior calculus.publishe