Perfect Gauge Actions on Anisotropic Lattices

On anisotropic lattices, where generally the lattice is rather coarse in spatial directions, a parametrized classically perfect action could help reducing lattice artifacts considerably. We investigate the possibility of constructing such actions for SU(3) gauge theory. We present two different methods to do so, either repeating the procedure used to create our newly parametrized isotropic FP action, or performing one single step starting with the isotropic result. The anisotropic action is parametrized using an ansatz including anisotropically APE-like smeared (``fat'') links. The parametrized classically perfect action with anisotropy $\xi=a_s/a_t=2$ is constructed and the renormalized anisotropy is measured using the torelon dispersion relation. It turns out that the renormalization is small.Comment: Lattice 2000 (Improvement and Renormalisation), 4 pages, 9 eps-figures, Late

(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods

We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse