An analysis of HDG methods for convection-dominated diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 225-256.

We present the first a priori error analysis of the h-version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection-dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L 2 -error of the scalar variable converges with order k+1/2 on general conforming quasi-uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L 2 -convergence order of k+1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results.

Reçu le :
DOI : 10.1051/m2an/2014032
Classification : 65N30
Mots-clés : HDG, convection-dominated diffusion
Fu, Guosheng 1 ; Qiu, Weifeng 2 ; Zhang, Wujun 3

1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, P.R. China.
3 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA.
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Fu, Guosheng; Qiu, Weifeng; Zhang, Wujun. An analysis of HDG methods for convection-dominated diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 225-256. doi : 10.1051/m2an/2014032. http://archive.numdam.org/articles/10.1051/m2an/2014032/

B. Ayuso and D. Marini, Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems. SIAM J. Numer. Anal. 47 (2009) 1391–1420. | DOI | MR | Zbl

C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311–341. | DOI | MR | Zbl

F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Russo and E. Süli, A Priori Error Analysis of Residual-freeBubbles for Advection-Diffusion Problems. SIAM J. Numer. Anal. 36 (1999) 1933–1948. | DOI | MR | Zbl

F. Brezzi, L.D. Marini and E. Süli, Residual-free bubbles for advection-diffusion problems: the general error analysis. Numer. Math. 85 (2000) 31–47. | DOI | MR | Zbl

A.N. Brooks and T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. | DOI | MR | Zbl

A. Buffa, T.J.R. Hughes and G. Sangalli, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44 (2006) 1420–1440. | DOI | MR | Zbl

Erik Burman and Alexandre Ern, Stabilized Galerkin Approximation of Convection-Diffusion-Reaction equations: Discrete Maximum Principle and Convergence. Math. Comput. 74 (2005) 1637–1652. | DOI | MR | Zbl

P. Castillo, B. Cockburn, D. Schötzau and C. Schwab, An optimal a priori error estimate for the hp–version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71 (2002) 455–478. | DOI | MR | Zbl

Y. Chen and B. Cockburn, Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: General nonconforming meshes. IMA J. Num. Anal. 32 (2012) 1267–1293. | DOI | MR | Zbl

Y. Chen and B. Cockburn, Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes. To appear in Math. Comput. | MR | Zbl

B. Cockburn, A Discontinuous Galerkin methods for convection-dominated problems, In High-order methods for computational physics, vol. 9 of Lect. Notes Comput. Sci. Eng. Springer, Berlin (1999) 69–224. | MR | Zbl

B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, In The mathematics of finite elements and applications, X, MAFELAP 1999 (Uxbridge). Elsevier, Oxford (2000) 225–238. | MR | Zbl

B. Cockburn and B. Dong, An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems. J. Sci. Comput. 32 (2007) 233–262. | DOI | MR | Zbl

B. Cockburn, B. Dong and J. Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77 (2008) 1887–1916. | DOI | MR | Zbl

B. Cockburn, B. Dong and J. Guzmán, Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 48 (2008) 1250–1265. | DOI | MR | Zbl

B. Cockburn, B. Dong, J. Guzmán, M. Restelli and R. Sacco, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. J. Sci. Comput. 31 (2009) 3827–3846. | MR | Zbl

B. Cockburn, B. Dong, J. Guzmán and J. Qian, Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity. SIAM J. Numer. Anal. 46 (2010) 133–146. | DOI | MR | Zbl

B. Cockburn, O. Dubois, J. Gopalakrishnan and S. Tan, Multigrid for an HDG method. To appear in IMA J. Numer. Anal. | MR | Zbl

B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. | DOI | MR | Zbl

B. Cockburn, J. Gopalakrishnan and F.-J. Sayas, A projection-based error analysis of HDG methods. Math. Comput. 79 (2010) 1351–1367. | DOI | MR | Zbl

B. Cockburn, W. Qiu and K. Shi, Conditions for superconvergence of HDG Methods for second-order elliptic problems. Math. Comput. 81 (2012) 1327–1353. | DOI | MR | Zbl

B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. | DOI | MR | Zbl

B. Cockburn and C.W. Shu, Runge–Kutta discontinuous Galerkin method for convection-dominated problems. J. Sci. Comput. 16 (2001) 173–261. | DOI | MR | Zbl

A. Devinatz, R. Ellis and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II. Indiana Univ. Math. J. 23 (1973-1974) 991–1011. | DOI | MR | Zbl

W. Eckhaus, Boundary layers in linear elliptic singular perturbation problems. SIAM Rev. 44 (1972). | MR | Zbl

H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection diffusion problems. IMA J. Num. Anal. 30 (2010) 1206–1234. | DOI | MR | Zbl

H. Goering, A. Felgenhauer, G. Lube, H.-G. Roos and L. Tobiska, Singularly perturbed differential equations. Akademie-Verlag, Berlin (1983). | MR | Zbl

J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527–550. | DOI | MR | Zbl

P. Houston, C. Schwab and E. Süli, Stabilized hp–finite element methods for first order hyperbolic problems. SIAM J. Numer. Anal. 37 (2000) 1618–1643. | DOI | MR | Zbl

P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133–2163. | DOI | MR | Zbl

T.J.R. Hughes, G. Scovazzi, P. Bochev and A. Buffa, A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2761–2787. | DOI | MR | Zbl

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46 (1986) 1–26. | DOI | MR | Zbl

T. Iliescu, A flow-aligning algorithm for convection-dominated problems. Int. J. Numer. Methods Engrg. 46 (1999) 993–1000. | DOI | MR | Zbl

T.E. Peterson, 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. Reed and T. Hill, Triangular mesh methods for the neutron transport equation. Technical Report LA- UR-73-479, Los Alamos Scientific Laboratory (1973).

N.C. Nguyen, J. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys. 288 (2009) 3232–3254. | DOI | MR | Zbl

N.C. Nguyen, J. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys. 288 (2009) 8841–8855. | DOI | MR | Zbl

H.-G. Roos, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008–2012. ISRN Appl. Math. (2012) 1–30. | MR | Zbl

H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. 2nd edition. In vol. 24 of Springer Series Comput. Math. Springer-Verlag, Berlin (2008). | MR | Zbl

H. Zarin and H.G. Roos, Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers. Numer. Math. 100 (2005) 735–759. | DOI | MR | Zbl

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