TY - GEN

T1 - The hybridizable discontinuous Galerkin methods

AU - Cockburn, Bernardo

PY - 2010/12/1

Y1 - 2010/12/1

N2 - In this paper, we present and discuss the so-called hybridizable discontinuous Galerkin (HDG) methods. The discontinuous Galerkin (DG) methods were originally devised for numerically solving linear and then nonlinear hyperbolic problems. Their success prompted their extension to the compressible Navier-Stokes equations - and hence to second-order elliptic equations. The clash between the DG methods and decades-old, well-established finite element methods resulted in the introduction of the HDG methods. The HDG methods can be implemented more efficiently and are more accurate than all previously known DG methods; they represent a competitive alternative to the well established finite element methods. Here we show how to devise and implement the HDG methods, argue why they work so well and prove optimal convergence properties in the framework of diffusion and incompressible flow problems. We end by briefly describing extensions to other continuum mechanics and fluid dynamics problems.

AB - In this paper, we present and discuss the so-called hybridizable discontinuous Galerkin (HDG) methods. The discontinuous Galerkin (DG) methods were originally devised for numerically solving linear and then nonlinear hyperbolic problems. Their success prompted their extension to the compressible Navier-Stokes equations - and hence to second-order elliptic equations. The clash between the DG methods and decades-old, well-established finite element methods resulted in the introduction of the HDG methods. The HDG methods can be implemented more efficiently and are more accurate than all previously known DG methods; they represent a competitive alternative to the well established finite element methods. Here we show how to devise and implement the HDG methods, argue why they work so well and prove optimal convergence properties in the framework of diffusion and incompressible flow problems. We end by briefly describing extensions to other continuum mechanics and fluid dynamics problems.

KW - Convection

KW - Diffusion

KW - Discontinuous Galerkin methods

KW - Finite element methods

KW - Incompressible fluid flow

KW - Mixed methods

UR - http://www.scopus.com/inward/record.url?scp=84877929518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877929518&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84877929518

SN - 9814324302

SN - 9789814324304

T3 - Proceedings of the International Congress of Mathematicians 2010, ICM 2010

SP - 2749

EP - 2775

BT - Proceedings of the International Congress of Mathematicians 2010, ICM 2010

T2 - International Congress of Mathematicians 2010, ICM 2010

Y2 - 19 August 2010 through 27 August 2010

ER -