Error analysis of the supg finite element discretization of evolutionary convection-diffusion-reaction equations

SIAM Journal on Numerical Analysis

Volumn 49, Issue 3, 2011, Pages 1149-1176

  John, Volker ^a,b   Novo, Julia ^c  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

^c UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

Backward Euler scheme; Crank Nicolson scheme; Error analysis; Evolutionary convection diffusion reaction equation; Streamline upwind Petrov Galerkin (SUPG) finite element method; Timecontinuous problem

Indexed keywords

BACKWARD EULER SCHEME; CRANK-NICOLSON SCHEME; EVOLUTIONARY CONVECTION-DIFFUSION-REACTION EQUATION; STREAMLINE-UPWIND PETROV-GALERKIN; TIMECONTINUOUS PROBLEM;

DIFFUSION IN LIQUIDS; ERROR ANALYSIS; ESTIMATION; GALERKIN METHODS; HEAT CONVECTION; INCOMPRESSIBLE FLOW; PARAMETER ESTIMATION; STABILIZATION;

FINITE ELEMENT METHOD;

EID: 79960435756     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/100789002     Document Type: Article

References (18)

1. P. B. Bochev, M. D. Gunzburger, and J. N. Shadid, Stability of the SUPG finite element method for transient advection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 193 (2004), pp. 2301-2323. (Pubitemid 38503358)
2. 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), pp. 199-259.
3. E. Burman, Consistent SUPG-method for transient transport problems: Stability and convergence, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1114-1123.
4. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.
5. P. G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis II, P. G. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, New York, Oxford, Tokyo, 1991, pp. 17-351.
6. R. Codina, Comparison of some finite element methods for solving the diffusion-convectionreaction equation, Comput. Methods Appl. Mech. Engrg., 156 (1998), pp. 185-210. (Pubitemid 128396484)
7. J. L. Guermond, Subgrid stabilization of Galerkin approximations of linear contraction semigroups of class C0 in Hilbert spaces, Numer. Methods Partial Differential Equations, 17 (2001), pp. 1-25.
8. M. C. Hsu, Y. Bazilevs, V. M. Calo, T. E. Tezduyar, and T. J. R. Hughes, Improving stability of stabilized and multiscale formulations in flow simulations at small time steps, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 828-840.
9. T. J. R. Hughes and A. N. Brooks, A multidimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows, AMD 34, T. J. R. Hughes, ed., ASME, New York, 1979, pp. 19-35.
10. T. J. R. Hughes, L. P. Franca, and M. Mallet, A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 63 (1987), pp. 97-112. (Pubitemid 17638984)
11. V. John, T. Mitkova, M. Roland, K. Sundmacher, L. Tobiska, and A. Voigt, Simulations of population balance systems with one internal coordinate using finite element methods, Chem. Engrg. Sci., 64 (2009), pp. 733-741.
12. V. John and E. Schmeyer, Stabilized finite element methods for time-dependent convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Engrg., 198 (2008), pp. 475-494.
13. V. John and E. Schmeyer, On finite element methods for 3d time-dependent convection-diffusion-reaction equations with small diffusion, in BAIL 2008-Boundary and Interior Layers, Lect. Notes Comput. Sci. Eng. 69, Springer, Berlin, 2009, pp. 173-182.
14. R. J. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33 (1996), pp. 627-665.
15. G. Lube and D. Weiss, Stabilized finite element methods for singularly perturbed parabolic problems, Appl. Numer. Math., 17 (1995), pp. 431-459.
16. H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed., Springer Ser. Comput. Math. 24, Springer, Berlin, 2008.
17. M. Stynes, Steady-state convection-diffusion problems, in Acta Numerica, A. Iserles, ed., Cambridge University Press, Cambridge, UK, 2005, pp. 445-508. (Pubitemid 40763887)
18. L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Math. 1605, Springer-Verlag, Berlin, 1975.