Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents for better readability, part on wavelet preconditioner adde

Cooperative Learning Approach to Delivering Professional Modules to Bachelor and Master Students: TPU Experience

Modern tendencies in engineering education set certain requirements on the competences to be obtained. To enable a future engineer to work at a highly competitive level and become an internationally recognized specialist, they must possess advanced English speaking and writing skills. At present most universities introduce educational programs that attract bright domestic and international students to ensure that they are in demand at the global market. The Strategic Program on Competitiveness Enhancement of National Research Tomsk Polytechnic University proposes that modules of professional training should be developed and delivered in English as a language of instruction so that the quality of teaching could be raised. To develop their own approaches to teaching, improve their language proficiency and develop courses and their elements in English, TPU teachers are to meet these challenging requirements and design course documentation and teaching aids, and engage modern teaching approaches in order to ensure high quality work with student groups which are often mixed in their abilities in both professional subjects and language proficiency levels. This paper addresses the course "Professional Training in English" which was developed upon completion of the joint program of TPU and the University of Southampton "Delivering through the Medium of English" and has successfully been delivered to bachelor students with the major "Oil and Gas Engineering" using the cooperative learning approach. The cooperative learning approach has proved to be one of the most successful in terms of teaching professional English to groups of students with mixed abilities

Real-time identification of the current density profile in the JET Tokamak: method and validation

International audienceThe real-time reconstruction of the plasma magnetic equilibrium in a Tokamak is a key point to access high performance regimes. Indeed, the shape of the plasma current density profile is a direct output of the reconstruction and has a leading effect for reaching a steady-state high performance regime of operation. In this paper we present the methodology followed to identify numerically the plasma current density in a Tokamak and its equilibrium. In order to meet the real-time requirements a C++ software has been developed using the combination of a finite element method, a nonlinear fixed point algorithm associated to a least square optimization procedure. The experimental measurements that enable the identification are the magnetics on the vacuum vessel, the interferometric and polarimetric measurements on several chords and the motional Stark effect. Details are given about the validation of the reconstruction on the JET tokamak, either by comparison with 'off-line' equilibrium codes or real time software computing global quantities