A continuous finite element method with face penalty to approximate friedrichs' systems

Mathematical Modelling and Numerical Analysis

Volumn 41, Issue 1, 2007, Pages 55-76

  Burman, Erik ^a   Ern, Alexandre ^b  

^a EPFL   (Switzerland)

^b UNIVERSITÉ PARIS EST   (France)

Author keywords

Finite elements; First order PDE's; Friedrichs' systems; Interior penalty; Stabilization methods

Indexed keywords

EID: 34547437817     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2007007     Document Type: Article

References (26)

1. I. Babuška, The finite element method with penalty. Math. Comp. 27 (1973) 221-228.
2. I. Babuška, and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863-875.
3. G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31 (1977) 45-59.
4. A. Bonito and E. Burman, A face penalty method for the three fields Stokes equation arising from Oldroyd-B viscoelastic flows, in Numerical Mathematics and Advanced Applications, ENUMATH Conf. Proc, Springer (2006).
5. E. Burman, A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43 (2005) 2012-2033.
6. E. Burman and P. Hansbo, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437-1453.
7. E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410.
8. E. Burman and B. Stamm, Discontinuous and continuous finite element methods with interior penalty for hyperbolic problems. J. Numer. Math (2005) Submitted (EPFL-IACS report 17.2005).
9. Z. Cai, T.A. Manteuffel, S.F. McCormick and S.V. Parter. First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem. SIAM J. Numer. Anal. 35 (1998) 320-335.
10. J. Douglas, Jr., and T. Dupont, Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lect. Notes Phys. 58, Springer-Verlag, Berlin (1976).
11. L. El Alaoui and A. Ern, Residual and hierarchical o posteriori estimates for nonconforming mixed finite element methods. ESAIM: M2AN 38 (2004) 903-929.
12. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Appl. Math. Sci. 159, Springer-Verlag, New York, NY (2004).
13. A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753-778.
14. A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order PDEs. SIAM J. Numer. Anal. 44 (2006) 2363-2388.
15. A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems. III. Multi-field theories with partial coercivity. SIAM J. Numer. Anal. (2006) Submitted (CERMICS report 2006-320).
16. A. Ern and J.-L. Guermond, Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM: M2AN 40 (2006) 29-48.
17. R.S. Falk and G.R. Richter, Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1999) 935-952.
18. K.O. Friedrichs, Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958) 333-418.
19. F. Hecht and O. Pironneau, FreeFEM++ Manual. Laboratoire Jacques-Louis Lions, University Paris VI (2005).
20. R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for non-conforming finite element methods. RAIRO Math. Model. Anal. Numer. 30 (1996) 237-263.
21. M. Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions. Ph.D. thesis, University of Oxford (2004).
22. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1-26.
23. O. Karakashian and F. Pascal, A-posteriori error estimates for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374-2399.
24. P. Lesaint, Finite element methods for symmetric hyperbolic equations. Numer. Math. 21 (1973/74) 244-255.
25. P. Lesaint, Sur la résolution des systèmes hyperboliques du premier ordre par des méthodes d'éléments finis. Ph.D. thesis, University of Paris VI, France (1975).
26. P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boors Ed., Academic Press (1974) 89-123.