Mathematiques Et Applications

2 primary works

Elements finis: theorie, applications, mise en oeuvre

by Alexandre Ern and Jean-Luc Guermond

Published 11 January 2002

Ces notes de cours (Ecole Nationale des Ponts et Chaussees, DEA de Mecanique de Paris VI) presentent la methode des elements finis dans un cadre mathematique rigoureux. En accordant une place fondamentale aux conditions inf-sup, elles s'affranchissent du cadre reducteur Lax-Milgram/Galerkin standard. Elles couvrent un spectre d'applications relativement large et apportent de nombreuses precisions sur la mise en oeuvre numerique. Trois plans de lecture sont proposes: le premier concu pour un lecteur interesse par les aspects mathematiques, le deuxieme s' adressant aux ingenieurs et le troisieme limite aux aspects elementaires. Les prerequis mathematiques, de niveau 2eme cycle universitaire, sont rappeles dans deux annexes.

Mathematical Aspects of Discontinuous Galerkin Methods

by Daniele Antonio Di Pietro and Alexandre Ern

Published 4 November 2011

This book introduces the basic ideas to build discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. The presentation is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite element and finite volume viewpoints are exploited to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed.