Hybridizable Discontinuous Galerkin Methods

We present an overview of recent developments of HDG methods for numerically solving partial differential equations in fluid mechanics.

[1]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations , 2011, J. Comput. Phys..

[2]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[3]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[4]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[5]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[6]  Haiying Wang,et al.  Superconvergent discontinuous Galerkin methods for second-order elliptic problems , 2009, Math. Comput..

[7]  Bernardo Cockburn,et al.  A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations , 2010, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition.

[8]  R. Stenberg A family of mixed finite elements for the elasticity problem , 1988 .

[9]  Rolf Stenberg,et al.  Postprocessing schemes for some mixed finite elements , 1991 .

[10]  Bernardo Cockburn,et al.  Error estimates for finite element methods for scalar conservation laws , 1996 .

[11]  Bernardo Cockburn,et al.  Discontinuous Galerkin Methods for Convection-Dominated Problems , 1999 .

[12]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[13]  Haiying Wang,et al.  Locally Conservative Fluxes for the Continuous Galerkin Method , 2007, SIAM J. Numer. Anal..

[14]  Bernardo Cockburn,et al.  A new elasticity element made for enforcing weak stress symmetry , 2010, Math. Comput..

[15]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

[16]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[17]  Per-Olof Persson,et al.  Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier--Stokes Equations , 2008, SIAM J. Sci. Comput..

[18]  Bernardo Cockburn,et al.  A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations , 2010 .

[19]  Bernardo Cockburn,et al.  An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[20]  Guido Kanschat,et al.  A locally conservative LDG method for the incompressible Navier-Stokes equations , 2004, Math. Comput..

[21]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[22]  J. Nédélec A new family of mixed finite elements in ℝ3 , 1986 .

[23]  Fatih Celiker,et al.  Hybridizable Discontinuous Galerkin Methods for Timoshenko Beams , 2010, J. Sci. Comput..

[24]  See-Chew Soon,et al.  Hybridizable discontinuous Galerkin method for solid mechanics , 2008 .

[25]  Bernardo Cockburn,et al.  journal homepage: www.elsevier.com/locate/cma , 2022 .

[26]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[27]  Gerard R. Richter,et al.  On the order of convergence of the discontinuous Galerkin method for hyperbolic equations , 2008, Math. Comput..

[28]  Sukru Guzey,et al.  The embedded discontinuous Galerkin method: application to linear shell problems , 2007 .

[29]  Bo Dong,et al.  A Hybridizable Discontinuous Galerkin Method for Steady-State Convection-Diffusion-Reaction Problems , 2009, SIAM J. Sci. Comput..

[30]  B. Rivière,et al.  Superconvergence and H(div) projection for discontinuous Galerkin methods , 2003 .

[31]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[32]  Bo Dong,et al.  Optimal Convergence of the Original DG Method for the Transport-Reaction Equation on Special Meshes , 2008, SIAM J. Numer. Anal..

[33]  Francisco-Javier Sayas,et al.  Analysis of HDG methods for Stokes flow , 2011, Math. Comput..

[34]  Todd E. Peterson,et al.  A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation , 1991 .

[35]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[36]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[37]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[38]  Bernardo Cockburn,et al.  A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems , 2004, SIAM J. Numer. Anal..

[39]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations , 2009, J. Comput. Phys..

[40]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[41]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[42]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[43]  Ricardo H. Nochetto,et al.  Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations , 1989 .

[44]  Francisco-Javier Sayas,et al.  A PROJECTION-BASED ERROR ANALYSIS OF HDG METHODS , 2010 .

[45]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[46]  Bo Dong,et al.  A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems , 2008, Math. Comput..

[47]  Bo Dong,et al.  A Hybridizable and Superconvergent Discontinuous Galerkin Method for Biharmonic Problems , 2009, J. Sci. Comput..

[48]  L. D. Marini,et al.  Stabilization mechanisms in discontinuous Galerkin finite element methods , 2006 .

[49]  Bernardo Cockburn,et al.  The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow , 2009, SIAM J. Numer. Anal..

[50]  Bernardo Cockburn,et al.  Devising discontinuous Galerkin methods for non-linear hyperbolic conversation laws , 2001 .

[51]  Bernardo Cockburn,et al.  A hybridizable discontinuous Galerkin method for linear elasticity , 2009 .

[52]  J. Nédélec Mixed finite elements in ℝ3 , 1980 .

[53]  Bernardo Cockburn,et al.  A Comparison of HDG Methods for Stokes Flow , 2010, J. Sci. Comput..