We consider the -version interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the advection-diffusion-reaction equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, new -version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration. The proposed method employs elemental polynomial bases of total degree (-basis) defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. Numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the -version DGFEM employing a -basis on both polytopic and tensor-product elements with a (standard) DGFEM employing a (mapped) -basis. Moreover, a computational example is also presented which demonstrates the performance of the proposed -version DGFEM on general agglomerated meshes.
DOI : 10.1051/m2an/2015059
Mots clés : Discontinuous Galerkin, polygonal elements, polyhedral elements, hp-finite element methods, inverse estimates, ��-basis, PDEs with nonnegative characteristic form
@article{M2AN_2016__50_3_699_0, author = {Cangiani, Andrea and Dong, Zhaonan and Georgoulis, Emmanuil H. and Houston, Paul}, title = {$hp${-Version} discontinuous {Galerkin} methods for advection-diffusion-reaction problems on polytopic meshes}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {699--725}, publisher = {EDP-Sciences}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/m2an/2015059}, mrnumber = {3507270}, zbl = {1342.65213}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015059/} }
TY - JOUR AU - Cangiani, Andrea AU - Dong, Zhaonan AU - Georgoulis, Emmanuil H. AU - Houston, Paul TI - $hp$-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 699 EP - 725 VL - 50 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015059/ DO - 10.1051/m2an/2015059 LA - en ID - M2AN_2016__50_3_699_0 ER -
%0 Journal Article %A Cangiani, Andrea %A Dong, Zhaonan %A Georgoulis, Emmanuil H. %A Houston, Paul %T $hp$-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 699-725 %V 50 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015059/ %R 10.1051/m2an/2015059 %G en %F M2AN_2016__50_3_699_0
Cangiani, Andrea; Dong, Zhaonan; Georgoulis, Emmanuil H.; Houston, Paul. $hp$-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 699-725. doi : 10.1051/m2an/2015059. http://www.numdam.org/articles/10.1051/m2an/2015059/
Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. M2AN 41 (2007) 21–54. | DOI | Numdam | MR | Zbl
and ,Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. M2AN 42 (2008) 443–469. | DOI | Numdam | MR | Zbl
and ,A class of domain decomposition preconditioners for -discontinuous Galerkin finite element methods. J. Sci. Comput. 46 (2011) 124–149. | DOI | MR | Zbl
and ,-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35 (2013) A1417–A1439. | DOI | MR | Zbl
, and ,Domain decomposition preconditioners for Discontinuous Galerkin methods for elliptic problems on complicated domains. J. Sci. Comput. 60 (2014) 203–227. | DOI | MR | Zbl
, and ,P.F. Antonietti, P. Houston, M. Sarti and M. Verani, Multigrid algorithms for -version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes. Preprint (2014). | arXiv | MR
Multigrid algorithms for -Discontinuous Galerkin discretizations of elliptic problems. SIAM J. Numer. Anal. 53 (2015) 598–618. | DOI | MR | Zbl
, and ,Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749–1779. | DOI | MR | Zbl
, , and ,Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47 (2009) 1391–1420. | DOI | MR | Zbl
and ,The finite element method with penalty. Math. Comput. 27 (1973) 221–228. | DOI | MR | Zbl
,The - version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987) 199–238. | DOI | Numdam | MR | Zbl
and ,The optimal convergence rate of the -version of the finite element method. SIAM J. Numer. Anal. 24 (1987) 750–776. | DOI | MR | Zbl
and ,Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31 (1977) 45–59. | DOI | MR | Zbl
,Agglomeration-based physical frame dG discretizations: An attempt to be mesh free. Math. Models Methods Appl. Sci. 24 (2014) 1495–1539. | DOI | MR | Zbl
, and ,On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231 (2012) 45–65. | DOI | MR | Zbl
, , , and ,Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations. Comput. Fluids 61 (2012) 77–85. | DOI | MR | Zbl
, , and ,Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2 (2005) 3–18. | DOI | MR | Zbl
and ,Multigrid methods for the symmetric interior penalty method on graded meshes. Numer. Linear Algebra Appl. 16 (2009) 481–501. | DOI | MR | Zbl
, and ,Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44 (2006) 1420–1440. | DOI | MR | Zbl
, and ,On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems. J. Sci. Comput. 57 (2013) 313–330. | DOI | MR | Zbl
, , and ,–version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24 (2014) 2009–2041. | DOI | MR | Zbl
, and ,Optimal convergence estimates for the trace of the polynomial -projection operator on a simplex. Math. Comput. 81 (2012) 765–787. | DOI | MR | Zbl
,P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 4 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl
B. Cockburn, An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems. In Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997). Springer, Berlin (1998) 151–268. | MR | Zbl
B. Cockburn, G.E. Karniadakis and C.-W. Shu., Eds., Discontinuous Galerkin Methods. Theory, Computation and Applications. Papers from the 1st International Symposium held in Newport, RI, May 24–26 1999. In Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2000). | MR | Zbl
Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46 (2008) 1250–1265. | DOI | MR | Zbl
, and ,Optimal convergence of the original DG method on special meshes for variable transport velocity. SIAM J. Numer. Anal. 48 (2010) 133–146. | DOI | MR | Zbl
, , and ,D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Math. Appl. Springer, Heidelberg (2012). | MR | Zbl
Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal., 39 (2001) 1343–1365. | DOI | MR | Zbl
and ,E.H. Georgoulis, Discontinuous Galerkin methods on shape-regular and anisotropic meshes. D. Phil. thesis, University of Oxford (2003).
Inverse-type estimates on -finite element spaces and applications. Math. Comput. 77 (2008) 201–219. | DOI | MR | Zbl
,A note on the design of -version interior penalty discontinuous Galerkin finite element methods for degenerate problems. IMA J. Numer. Anal. 26 (2006) 381–390. | DOI | MR | Zbl
and ,-Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains. Num. Meth. Partial Differ. Eqs. 30 (2014) 1342–1367. | DOI | MR | Zbl
and ,Stabilized -finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal. 37 (2000) 1618–1643. | DOI | MR | Zbl
, and ,Stabilised -finite element approximation of partial differential equations with nonnegative characteristic form. Computing 66 (2001) 99–119. | DOI | MR | Zbl
and ,Discontinuous -finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133–2163. | DOI | MR | Zbl
, and ,An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46 (1986) 1–26. | DOI | MR | Zbl
and ,A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1999) 359–392. | DOI | MR | Zbl
and ,An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comput. 72 (2003) 1215–1238. | DOI | MR | Zbl
and ,K. Lipnikov, D. Vassilev and I. Yotov, Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes-Darcy flows on polygonal and polyhedral grids. Numer. Math. (2013) 1–40. | MR | Zbl
Polynomial liftings on a tetrahedron and applications to the -version of the finite element method in three dimensions. SIAM J. Numer. Anal. 34 (1997) 282–314. | DOI | MR | Zbl
,Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Uni. Hamburg 36 (1971) 9–15. | DOI | MR | Zbl
,An -analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17 (2002) 561–571. | DOI | MR | Zbl
and ,A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28 (1991) 133–140. | DOI | MR | Zbl
,W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973).
C. Schwab, – and –Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Oxford University Press (1998). | MR | Zbl
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton, University Press, Princeton, N.J. (1970). | MR | Zbl
Polymesher: A general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidisc. Optim. 45 (2012) 309–328,. | DOI | MR | Zbl
, , and ,R. Verfürth, On the constants in some inverse inequalities for finite element functions. Technical Report 257, University of Bochum (1999).
Discontinuous Galerkin methods with nodal and hybrid modal/nodal triangular, quadrilateral, and polygonal elements for nonlinear shallow water flow. Comput. Methods Appl. Mech. Engrg. 270 (2014) 113–149. | DOI | MR | Zbl
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