Degré d'un polynôme

Résumé

En algèbre commutative, le degré d'un polynôme (en une ou plusieurs indéterminées) est le degré le plus élevé de ses termes lorsque le polynôme est exprimé sous sa forme canonique constituée d'une somme de monômes. Le degré d'un terme est la somme des exposants des indéterminées qui y apparaissent. Le terme ordre a été utilisé comme synonyme de degré, mais de nos jours, il fait référence à des concepts différents, bien que connexes. Par exemple, le polynôme 7XY + 4X – 9 a trois monômes. Le premier est de degré 2 + 3 = 5, le deuxième (4XY) de degré 1, et le dernier (–9XY) de degré 0. Par conséquent, le polynôme est de degré 5, qui est le degré le plus élevé de tous ses monômes. Pour déterminer le degré d'un polynôme qui n'est pas sous forme standard — par exemple (X + 1) – (X – 1) — on doit d'abord le mettre sous forme standard en développant les produits (par distributivité) et en combinant les termes semblables ; par exemple, (X + 1) – (X – 1) = 4X, et son degré est 1, bien que cha

Source officielle

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This work focuses on the development of a non-conforming method for the coupling of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a finite number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which include also the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains. In particular, our approach can be regarded as a specialization of the three-field method in which the spaces used to enforce the continuity of the solution and its conormal derivative across the interface are taken equal. One of the possible choices to approximate the interface variational space – which we consider here – is by Fourier basis functions. As we show in the numerical simulations, the method is well-suited for the coupling of problems defined on globally non-conforming meshes or discretized with basis functions of different polynomial degree in each subdomain. We also investigate the possibility of coupling solutions obtained with incompatible numerical methods, namely the finite element method and isogeometric analysis.

2019

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We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation k >= 0. For face unknowns that are polynomials of degree k, we devise two versions of the method, depending on the polynomial degree (k - 1) or k of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form (epsilon(1/2) + Hk+1 + (epsilon/H)(1/2)), and we illustrate our theoretical findings on some test-cases.

WALTER DE GRUYTER GMBH2019

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Computational Performance of a Parallelized Three-Dimensional High-Order Spectral Element Toolbox

Christoph Bosshard, Roland Bouffanais, Christian Clémençon, Michel Deville, Nicolas Fietier, Sohrab Jairo Kehtari, Vincent Keller, Jonas Marcel Lätt

In this paper, a comprehensive performance review of all MPI-based high-order three-dimensional spectral element method C++ toolbox is presented. The focus is put on the performance evaluation of aspects with a particular emphasis on the parallel efficiency. The performance evaluation is analyzed with help of a time prediction model based on a parameterization of the application and the hardware resources. A tailor-made CFD computation benchmark case is introduced and used to carry out this review, stressing the particular interest for clusters with up to 8192 cores. Some problems in the parallel implementation have been detected and corrected. The theoretical complexities with respect, to the number of elements, to the polynomial degree, and to communication needs are correctly reproduced. It is concluded that this type of code has a nearly perfect speed up on machines with thousands of crores, and is ready to make the step to next-generation petaflop machines

Springer-Verlag New York, Ms Ingrid Cunningham, 175 Fifth Ave, New York, Ny 10010 Usa2009

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