Efficiency of high‐order elements for continuous and discontinuous Galerkin methods

To evaluate the computational performance of high‐order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear versus high‐order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution. Continuous and discontinuous Galerkin methods are considered in two‐dimensional and three‐dimensional domains for simplices and parallelotopes. Moreover, both element‐wise and global operations arising from different Galerkin approaches are studied. The operation count estimates show, that for implicit solvers, high‐order methods are more efficient than linear ones. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Iain S. Duff.  Analysis of sparse systems , 1972 .

[2]  A. George.  Nested Dissection of a Regular Finite Element Mesh , 1973 .

[3]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[4]  O. Cessenat,et al.  Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem , 1998 .

[5]  Jean-François Remacle,et al.  An algorithm oriented mesh database , 2003, IMR.

[6]  C. Farhat,et al.  The Discontinuous Enrichment Method , 2000 .

[7]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[8]  Per-Olof Persson,et al.  The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems , 2007, SIAM J. Sci. Comput..

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

[10]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[11]  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..

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

[13]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations , 2009, Journal of Computational Physics.

[14]  Robert Michael Kirby,et al.  From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations , 2010, J. Comput. Phys..

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

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

[17]  Xevi Roca,et al.  GPU-accelerated sparse matrix-vector product for a hybridizable discontinuous Galerkin method , 2011 .

[18]  Rainald Löhner.  Error and work estimates for high‐order elements , 2011 .

[19]  S. Sherwin,et al.  From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements , 2011 .

[20]  Spencer J. Sherwin,et al.  From h to p Efficiently: Selecting the Optimal Spectral/hp Discretisation in Three Dimensions , 2011 .

[21]  Robert Michael Kirby,et al.  To CG or to HDG: A Comparative Study , 2012, J. Sci. Comput..

[22]  C. Farhat,et al.  Overview of the discontinuous enrichment method, the ultra‐weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons , 2012 .

[23]  Antonio Huerta,et al.  Are High-order and Hybridizable Discontinuous Galerkin methods competitive ? , 2012 .

[24]  Rémi Abgrall,et al.  High‐order CFD methods: current status and perspective , 2013 .

[25]  Ralf Hartmann,et al.  Higher‐order and adaptive discontinuous Galerkin methods with shock‐capturing applied to transonic turbulent delta wing flow , 2013 .

[26]  Va,et al.  Improved error and work estimates for high-order elements , 2013 .

[27]  Antonio Huerta,et al.  Hybridizable discontinuous Galerkin p‐adaptivity for wave propagation problems , 2013 .

[28]  A. Huerta,et al.  High‐order continuous and discontinuous Galerkin methods for wave problems , 2013 .