Energy norm shape a posteriori error estimation for mixed discontinuous Galerkin approximations of the stokes problem

Journal of Scientific Computing

Volumn 22-23, Issue , 2005, Pages 347-370

  Paul, Houston ^a   Schötzau, Dominik ^b   Wihler, Thomas P ^c  

^a UNIVERSITY OF LEICESTER   (United Kingdom)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^c University of Minnesota   (United States)

Author keywords

A posteriori errorestimation; Discontinuous Galerkin methods; Stokes problem

Indexed keywords

APPROXIMATION THEORY; ERROR ANALYSIS; FINITE ELEMENT METHOD; INCOMPRESSIBLE FLOW; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS; PROBLEM SOLVING;

A POSTERIORI ERROR ESTIMATION; DISCONTINUOUS GALERKIN METHOD; INCOMPRESSIBLE FLOW MODELS; STROKES PROBLEM;

GALERKIN METHODS;

EID: 17444413054     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-004-4143-7     Document Type: Article

References (24)

1. Arnold, D.N., Brezzi, F., Cockburn, B., and Marini, L. D. (2001). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749-1779.
2. Baker, G. A., Jureidini, W. N., and Karakashian, O. A. (1990). Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Numer. Anal. 27, 1466-1485.
3. Becker, R., Hansbo, P., and Larson, M. G. (2003). Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Meth. Appl. Mech. Eng. 192, 723-733.
4. Becker, R., Hansbo, P., and Stenberg, R. (2003). A finite element method for domain decomposition with non-matching grids. Math. Model. Anal. Numer. 37, 209-225.
5. Brezzi, F., and Fortin, M. (1991). Mixed and hybrid finite element methods. In Springer Series in Computational Mathematics, Vol. 15. Springer-Verlag, New York.
6. Cockburn, B., Kanschat, G., and Schötzau, D. (2004). The local discontinuous Galerkin method for the Oseen equations. Math. Comp. 73, 569-593.
7. Cockburn, B., Kanschat, G., Schötzau, D., and Schwab, C. (2002). Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40, 319-343.
8. Dauge, M. (1989). Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations. SIAM J. Math. Anal. 20(1), 74-97.
9. Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. (1995). Introduction to adaptive methods for differential equations. In Iserles, A. (ed.), Acta Numerica, Cambridge University Press, pp. 105-158.
10. Girault, V., and Raviart, P. A. (1986). Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, New York.
11. Girault, V., Rivière, B., and Wheeler, M. F. (2002). A Discontinuous Galerkin Method with Non-overlapping Domain Decomposition for the Stokes and Navier-Stokes problems, Technical Report 02-08, TICAM, UT Austin. In press in Math. Comp.
12. Hansbo, P., and Larson, M. G. (2002). Discontinuous finite element methods for incompressible and nearly incompressible elasticity by use of Nitsche's method. Comput. Meth. Appl. Mech. Eng. 191, 1895-1908.
13. Houston, P., Perugia, I., and Schötzau, D. (2004). Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J: Numer. Anal. 42, 434-459.
14. Houston, P., Perugia, I., and Schötzau, D. (2003). Energy Norm A Posteriori Error Estimation for Mixed Discontinuous Galerkin Approximations of the Maxwell Operator, Technical Report 2003-17, Department of Mathematics and Computer Science, University of Leicester.
15. Kanschat G., and Rannacher, R. (2002). Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems. J. Numer. Math. 10, 249-274.
16. Karakashian, O. A., and Jureidini, W. N. (1998). A nonconforming finite element method for the stationary Navier-Stokes equations. SIAM J. Numer. Anal. 35, 93-120.
17. Karakashian, O. A., and Pascal, F. (2003). A posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41, 2374-2399.
18. Perugia, I., and Schötzau, D. (2002). An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17, 561-571.
19. Rivière, B., and Wheeler. M. F. (2003). A Posteriori error estimates for a discontinous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46, 141-163.
20. Schötzau, D., Schwab, C., and Toselli, A. (2003). Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171-2194.
21. Schötzau, D., and Wihler, T. P. (2003). Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons. Numer. Math. 96, 339-361.
22. Toselli, A. (2002). hp-Discontinuous Galerkin approximations for the Stokes problem. Math. Models Meth. Appl. Sci. 12, 1565-1616.
23. Verfürth, R. (1996). A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner, Stuttgart.
24. Wihler, T. P., Frauenfelder, P., and Schwab, C. (2003). Exponential convergence of the hp-DGFEM for diffusion problems. Comput. Math. Appl. 46, 183-205.