A priori diffusion-uniform error estimates for nonlinear singularly perturbed problems: BDF2, midpoint and time DG
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 537-563.

This work deals with a nonlinear nonstationary semilinear singularly perturbed convection-diffusion problem. We discretize this problem by the discontinuous Galerkin method in space and by the midpoint rule, BDF2 and quadrature variant of discontinuous Galerkin in time. We present a priori error estimates for these three schemes that are uniform with respect to the diffusion coefficient going to zero and valid even in the purely convective case.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016035
Classification : 65M12, 65M15, 65M60
Mots clés : Discontinuous Galerkin method, a priori error estimates, nonlinear convection-diffusion equation, diffusion-uniform error estimates
Kučera, Václav 1 ; Vlasák, Miloslav 1

1 Charles University in Prague, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Sokolovská 83, 18675 Prague 8, Czech Republic.
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     title = {A priori diffusion-uniform error estimates for nonlinear singularly perturbed problems: {BDF2,} midpoint and time {DG}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {537--563},
     publisher = {EDP-Sciences},
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Kučera, Václav; Vlasák, Miloslav. A priori diffusion-uniform error estimates for nonlinear singularly perturbed problems: BDF2, midpoint and time DG. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 537-563. doi : 10.1051/m2an/2016035. http://www.numdam.org/articles/10.1051/m2an/2016035/

G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. | DOI | Numdam | MR | Zbl

T. Barth and M. Ohlberger, Finite Volume Methods: Foundation and Analysis. Vol. 1 of Encyclopedia of Computational Mechanics. John Wiley & Sons, Chichester, New York, Brisbane (2004) 439–474.

F. Bassi and S. Rebay, High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations. J. Comput. Phys. 138 (1997) 251–285. | DOI | MR | Zbl

P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, Amsterdam (1978). | 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

V. Dolejší and M. Feistauer, Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow. Springer (2015). | MR

V. Dolejší and M. Vlasák, Analysis of a BDF-DG scheme for nonlinear convection-diffusion problems. Numer. Math. 110 (2008) 405–447. | DOI | MR | Zbl

V. Dolejší, M. Feistauer and V. Sobotíková, Analysis of the discontinuous Galerkin method for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 2709–2733. | DOI | MR | Zbl

V. Dolejší, M. Feistauer and J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2813–2827. | DOI | MR | Zbl

V. Dolejší, M. Feistauer, V. Kučera and V. Sobotíková, An optimal L (L 2 )-error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection-diffusion problem. IMA J. Numer. Anal. 28 (2008) 496–521. | DOI | MR | Zbl

M. Feistauer and V. Kučera, On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224 (2007) 208–221. | DOI | MR | Zbl

M. Feistauer, J. Hájek and K. Švadlenka, Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. Appl. Math. 52 (2007) 197–233. | DOI | MR | Zbl

S. Gianni, D. Schötzau and L. Zhu, An a posteriori error estimate for hp-adaptive DG methods for convection-diffusion problems on anisotropically refined meshes. Comp. Math. Appl. 67 (2014) 869–887. | DOI | MR | Zbl

E. Hairer, S.P. Norsett and G. Wanner, Solving ordinary differential equations I, Nonstiff problems. Springer Verlag (2000). | MR | Zbl

V. Kučera, Optimal L (L 2 )-error Estimates for the DG Method Applied to Nonlinear Convection-Diffusion Problems with Nonlinear Diffusion. Numer. Func. Anal. Opt. 31 (2010) 285–312. | DOI | MR | Zbl

V. Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems. IMA J. Numer. Anal. 32 (2014) 820–861. | MR | Zbl

V. Kučera, Finite element error estimates for nonlinear convective problems. J. Numer. Math. 24 (2016) 143–165. | DOI | MR | Zbl

S. Osher, Riemann solvers, the entropy condition, and difference approximations. SIAM. J. Numer. Anal. 21 (1984) 217–235. | 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).

M. Vlasák, Optimal spatial error estimates for DG time discretizations. J. Numer. Math. 21 (2013) 201–230. | DOI | MR | Zbl

M. Vlasák and H.G. Roos, An optimal uniform a priori error estimate for an unsteady singularly perturbed problem. Int. J. Numer. Anal. Model. 11 (2014) 24–33. | MR | Zbl

M. Vlasák, V. Dolejší and J. Hájek, A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Eqs. 27 (2011) 1456–1482. | DOI | MR | Zbl

E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators. Springer, Heidelberg (1986). | Zbl

Q. Zhang and C.W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws. SIAM J. Numer. Anal. 44 (2006) 1703–1720. | DOI | MR | Zbl

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