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.
Accepté le :
DOI : 10.1051/m2an/2016035
Mots-clés : Discontinuous Galerkin method, a priori error estimates, nonlinear convection-diffusion equation, diffusion-uniform error estimates
@article{M2AN_2017__51_2_537_0,
author = {Ku\v{c}era, V\'aclav and Vlas\'ak, Miloslav},
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},
volume = {51},
number = {2},
year = {2017},
doi = {10.1051/m2an/2016035},
mrnumber = {3626410},
zbl = {1372.65257},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016035/}
}
TY - JOUR AU - Kučera, Václav AU - Vlasák, Miloslav TI - A priori diffusion-uniform error estimates for nonlinear singularly perturbed problems: BDF2, midpoint and time DG JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 537 EP - 563 VL - 51 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016035/ DO - 10.1051/m2an/2016035 LA - en ID - M2AN_2017__51_2_537_0 ER -
%0 Journal Article %A Kučera, Václav %A Vlasák, Miloslav %T A priori diffusion-uniform error estimates for nonlinear singularly perturbed problems: BDF2, midpoint and time DG %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 537-563 %V 51 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016035/ %R 10.1051/m2an/2016035 %G en %F M2AN_2017__51_2_537_0
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://archive.numdam.org/articles/10.1051/m2an/2016035/
and , 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.
and , 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
and , 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
and , Analysis of a BDF-DG scheme for nonlinear convection-diffusion problems. Numer. Math. 110 (2008) 405–447. | DOI | MR | Zbl
, and , Analysis of the discontinuous Galerkin method for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 2709–2733. | DOI | MR | Zbl
, and , Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2813–2827. | DOI | MR | Zbl
, , and , An optimal -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
and , On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224 (2007) 208–221. | DOI | MR | Zbl
, and , Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. Appl. Math. 52 (2007) 197–233. | DOI | MR | Zbl
, and , 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
, Optimal -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
, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems. IMA J. Numer. Anal. 32 (2014) 820–861. | MR | Zbl
, Finite element error estimates for nonlinear convective problems. J. Numer. Math. 24 (2016) 143–165. | DOI | MR | Zbl
, 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).
, Optimal spatial error estimates for DG time discretizations. J. Numer. Math. 21 (2013) 201–230. | DOI | MR | Zbl
and , An optimal uniform a priori error estimate for an unsteady singularly perturbed problem. Int. J. Numer. Anal. Model. 11 (2014) 24–33. | MR | Zbl
, and , 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
and , 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
Cité par Sources :
