Numerical Studies of Galerkin-type Time-discretizations Applied to Transient Convection-diffusion-reaction Equations

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numerical results which shows that the cGP(k) and dG(k)- methods are accurate of order k +1, respectively, in the whole time interval. Moreover, the cGP(k)-method is superconvergent of order 2k and dG(k)-method is of order 2k +1 at the discrete time points. Furthermore, the dependence of the results on the choice of the stabilization parameter are discussed and compared.

• Naveed Ahmed
• Gunar Matthies

• Convection-diffusion-reaction equations
• stabilized finite elements
• discontinuous Galerkin
• continuous Galerkin-Petrov.

[1] N. Ahmed, G. Matthies, L. Tobiska, and H. Xie. Discontinuous Galerkin
time stepping with local projection stabilization for transient convectiondiffusion-
reaction problems. Comput. Methods Appl. Mech. Engrg.,
200(21-22):1747-1756, 2011.
[2] M. I. Asensio, B. Ayuso, and G. Sangalli. Coupling stabilized finite
element methods with finite difference time integration for advectiondiffusion-
reaction problems. Comput. Methods Appl. Mech. Engrg.,
196(35-36):3475-3491, 2007.
[3] A. K. Aziz and P. Monk. Continuous finite elements in space and time
for the heat equation. Math. Comp., 52(186):255-274, 1989.
[4] E. Burman and M. A. Fern'andez. Finite element methods with
symmetric stabilization for the transient convection-diffusion-reaction
equation. Comput. Methods Appl. Mech. Engrg., 198(33-36):2508-2519,
2009.
[5] E. Burman and P. Hansbo. The edge stabilization method for finite
elements in CFD. In Numerical mathematics and advanced applications,
pages 196-203. Springer, Berlin, 2004.
[6] P. G. Ciarlet. The finite element method for elliptic problems. North-
Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and
its Applications, Vol. 4.
[7] R. Codina. Comparison of some finite element methods for solving the
diffusion-convection-reaction equation. Comput. Methods Appl. Mech.
Engrg., 156(1-4):185-210, 1998.
[8] R. Codina. Stabilization of incompressibility and convection through
orthogonal sub-scales in finite element methods. Comput. Methods Appl.
Mech. Engrg., 190(13-14):1579-1599, 2000.
[9] R. Codina and J. Blasco. Analysis of a stabilized finite element
approximation of the transient convection-diffusion-reaction equation
using orthogonal subscales. Comput. Vis. Sci., 4(3):167-174, 2002.
[10] Miloslav Feistauer, Jaroslav H'ajek, and Karel Svadlenka. Space-time
discontinuous Galerkin method for solving nonstationary convectiondiffusion-
reaction problems. Appl. Math., 52(3):197-233, 2007.
[11] E. Hairer and G. Wanner. Solving ordinary differential equations. II,
volume 14 of Springer Series in Computational Mathematics. Springer-
Verlag, Berlin, second edition, 1996. Stiff and differential-algebraic
problems.
[12] L. He and L. Tobiska. The two-level local projection stabilization
as an enriched one-level approach. Adv. Comput. Math., 2011. DOI
10.1007/s10444-011-9188-1.
[13] T. J. R. Hughes and A. N. Brooks. A multidimensional upwind scheme
with no crosswind diffusion. In Finite element methods for convection
dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs.,
New York, 1979), volume 34 of AMD, pages 19-35. Amer. Soc. Mech.
Engrs. (ASME), New York, 1979.
[14] S. Hussain, F. Schieweck, and S. Turek. Higher order Galerkin time
discretizations and fast multigrid solvers for the heat equation. J. Numer.
Math, 19(1):41-61, 2011.
[15] V. John and G. Matthies. MooNMDÔÇöa program package based on
mapped finite element methods. Comput. Vis. Sci., 6(2-3):163-169,
2004.
[16] V. John and J. Novo. Error analysis of the SUPG finite element
discretization of evolutionary convection-diffusion-reaction equations.
SIAM J. Numer. Anal., 49(3):1149-1176, 2011.
[17] V. John and E. Schmeyer. Finite element methods for time-dependent
convection-diffusion-reaction equations with small diffusion. Comput.
Methods Appl. Mech. Engrg., 198(3-4):475-494, 2008.
[18] P. Knobloch. On the application of local projection methods to
convection-diffusion-reaction problems. In BAIL 2008ÔÇöboundary and
interior layers, volume 69 of Lect. Notes Comput. Sci. Eng., pages 183-
194. Springer, Berlin, 2009.
[19] P. Knobloch. A generalization of the local projection stabilization
for convection-diffusion-reaction equations. SIAM J. Numer. Anal.,
48(2):659-680, 2010.
[20] G. Lube and D. Weiss. Stabilized finite element methods for singularly
perturbed parabolic problems. Appl. Numer. Math., 17(4):431-459,
1995.
[21] G. Matthies and F. Schieweck. Higher order variational time
discretizations for nonlinear systems of ordinary differential equations.
Preprint 23/2011, Fakult¨at f¨ur Mathematik, Otto-von-Guericke-
Universit¨at Magdeburg, 2011.
[22] G. Matthies, P. Skrzypacz, and L. Tobiska. A unified convergence
analysis for local projection stabilisations applied to the Oseen problem.
M2AN Math. Model. Numer. Anal., 41(4):713-742, 2007.
[23] H. G. Roos, M. Stynes, and L. Tobiska. Robust numerical methods
for singularly perturbed differential equations, volume 24 of Springer
Series in Computational Mathematics. Springer-Verlag, Berlin, second
edition, 2008. Convection-diffusion-reaction and flow problems.
[24] F. Schieweck. A-stable discontinuous Galerkin-Petrov time discretization
of higher order. J. Numer. Math., 18(1):25-57, 2010.
[25] V. Thom'ee. Galerkin finite element methods for parabolic problems,
volume 25 of Springer Series in Computational Mathematics. Springer-
Verlag, Berlin, second edition, 2006.