FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part III: Hybridized discontinuous Galerkin (HDG) formulation

Abstract The third paper in our series on open source MATLAB/GNU Octave implementation of the discontinuous Galerkin (DG) method(s) focuses on a hybridized formulation. The main aim of this ongoing work is to develop rapid prototyping techniques covering a range of standard DG methodologies and suitable for small to medium sized applications. Our FESTUNG package relies on fully vectorized matrix/vector operations throughout, and all details of the implementation are fully documented. Once again, great care is taken to maintain a direct mapping between discretization terms and code routines as well as to ensure full compatibility to GNU Octave. The current work formulates a hybridized DG scheme for a linear advection problem, describes hybrid approximation spaces on the mesh skeleton, and compares the performance of this discretization to the standard (element-based) DG method for different polynomial orders.

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

[2]  D. Kröner.  Numerical Schemes for Conservation Laws , 1997 .

[3]  P. Tesini,et al.  High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations , 2009 .

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

[5]  Claus-Dieter Munz,et al.  Parallel Performance of a Discontinuous Galerkin Spectral Element Method Based PIC-DSMC Solver , 2015 .

[6]  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..

[7]  Balthasar Reuter,et al.  FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part I: Diffusion operator , 2014, Comput. Math. Appl..

[8]  Bernardo Cockburn,et al.  Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations , 2011, J. Comput. Phys..

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

[10]  Alexander Jaust,et al.  An efficient linear solver for the hybridized discontinuous Galerkin method , 2016 .

[11]  Herbert Egger,et al.  A hybrid mixed discontinuous Galerkin finite-element method for convection–diffusion problems , 2010 .

[12]  Chi-Wang Shu,et al.  Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments , 2014 .

[13]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

[14]  P. Tesini,et al.  On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations , 2012, J. Comput. Phys..

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

[16]  Béatrice Rivière,et al.  Inexact hierarchical scale separation: A two-scale approach for linear systems from discontinuous Galerkin discretizations , 2017, Comput. Math. Appl..

[17]  Ngoc Cuong Nguyen,et al.  Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics , 2012, J. Comput. Phys..

[18]  Jochen Schütz,et al.  A hybrid mixed method for the compressible Navier-Stokes equations , 2013, J. Comput. Phys..

[19]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[20]  Balthasar Reuter,et al.  FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part II: Advection operator and slope limiting , 2016, Comput. Math. Appl..

[21]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[22]  Santiago Badia,et al.  Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes , 2016, ArXiv.

[23]  Bernardo Cockburn,et al.  High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics , 2011, J. Comput. Phys..

[24]  Michael Woopen,et al.  A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow , 2013, ArXiv.

[25]  D. Arnold,et al.  Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates , 1985 .

[26]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .

[27]  Jochen Schütz,et al.  A hierarchical scale separation approach for the hybridized discontinuous Galerkin method , 2017, J. Comput. Appl. Math..

[28]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

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

[30]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

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

[32]  Alexander Jaust,et al.  A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows , 2014, 1406.0314.

[33]  Dmitri Kuzmin,et al.  Scale separation in fast hierarchical solvers for discontinuous Galerkin methods , 2015, Appl. Math. Comput..

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

[35]  David L. Darmofal,et al.  p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2005 .

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

[37]  Herbert Egger,et al.  A Hybrid Discontinuous Galerkin Method for Darcy-Stokes Problems , 2013, Domain Decomposition Methods in Science and Engineering XX.

[38]  Robert Michael Kirby,et al.  To CG or to HDG: A Comparative Study in 3D , 2016, J. Sci. Comput..

[39]  Balthasar Reuter,et al.  FESTUNG: A MATLAB /GNU Octave toolbox for the discontinuous Galerkin method. Part IV: Generic problem framework and model-coupling interface , 2018, ArXiv.

[40]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..

[41]  Luming Wang,et al.  A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations , 2015 .

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

[43]  Rainald Löhner,et al.  A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids , 2006 .

[44]  Vadym Aizinger,et al.  A Geometry Independent Slope Limiter for the Discontinuous Galerkin Method , 2011 .

[45]  Tan Bui-Thanh,et al.  From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations , 2015, J. Comput. Phys..

[46]  H. Egger,et al.  hp analysis of a hybrid DG method for Stokes flow , 2013 .

[47]  Dmitri Kuzmin,et al.  A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods , 2010, J. Comput. Appl. Math..