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Publication# Sur la méthode de Buser-Silhol pour l'uniformisation des surfaces de Riemann hyperelliptiques

Abstract

The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

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Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and po

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Rieman

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We investigate low-frequency electromagnetic wave propagation and absorption properties in 2D and 3D plasma configurations. For these purposes, we have developed a new full-wave 3D code LEMan that determines a global solution of the wave equation in bounded stellarator plasmas excited with an external antenna. No assumption on the wavelength compared to the plasma size is made, all the effects of the 3D geometry and finite plasma extent are included. The equation is formulated in terms of electromagnetic potentials in order to avoid numerical pollution effects. The code utilises linear and Hermite cubic finite element discretisation in the radial direction and Fourier series in the poloidal and toroidal variables. The full cold plasma model including finite electron inertia and, thus, mode conversion effects is implemented. The code uses Boozer magnetic coordinates and has an interface to the TERPSICHORE code. Special care is taken to treat the magnetic axis and to ensure the unicity of the numerical solution. The discretisation, interpolation and numerical derivation methods specifically adapted for our problem avoid the energy sink in the origin and provide a very good local and global energy conservation. A special algorithm has been developed to analytically expand the wave equation coefficients in the full 3D stellarator geometry. The code has been specifically optimised for vector computing platform, reaching close to maximum average performances on the NEC SX5 machine. The code has been applied in 1D, 2D, and 3D geometries. No unphysical solutions have been observed. LEMan successfully recovers all the fundamental properties of the Alfvén spectrum (gaps, eigenmodes). Benchmarks have been made against the 2D LION code and JET experimental measurements, showing a good agreement between the results.

The authors propose a numerical method for the uniformization of Riemann surfaces and algebraic curves in genus two with highly accurate results. Let $G$ be a Fuchsian group acting on the unit disk $Bbb D$, and let $S = Bbb D / G$. It is well known that $S$ is also in natural way an algebraic curve. The authors describe a practical way to compute, in genus 2, the uniformizing function from the unit disk to an algebraic curve. The basic idea underlying the method is to reduce the problem to the known case of genus 1. Moreover, they produce an algorithm to compute an approximation of uniformizing functions (the algorithm code and results of computations can be found at the second named author's homepage.

In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the Yosida-q methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example.