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

Erik Burman; Alexandre Ern

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 1, page 55-76
  • ISSN: 0764-583X

Abstract

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A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number of nonzero entries in the stiffness matrix is also proposed and analyzed. Finally, numerical results are presented to illustrate the theoretical analysis.

How to cite

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Burman, Erik, and Ern, Alexandre. "A continuous finite element method with face penalty to approximate Friedrichs' systems." ESAIM: Mathematical Modelling and Numerical Analysis 41.1 (2007): 55-76. <http://eudml.org/doc/250077>.

@article{Burman2007,
abstract = { A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number of nonzero entries in the stiffness matrix is also proposed and analyzed. Finally, numerical results are presented to illustrate the theoretical analysis. },
author = {Burman, Erik, Ern, Alexandre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite elements; interior penalty; stabilization methods; Friedrichs' systems; first-order PDE's.; finite elements; first-order PDEs; numerical results; convergence},
language = {eng},
month = {4},
number = {1},
pages = {55-76},
publisher = {EDP Sciences},
title = {A continuous finite element method with face penalty to approximate Friedrichs' systems},
url = {http://eudml.org/doc/250077},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Burman, Erik
AU - Ern, Alexandre
TI - A continuous finite element method with face penalty to approximate Friedrichs' systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/4//
PB - EDP Sciences
VL - 41
IS - 1
SP - 55
EP - 76
AB - A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number of nonzero entries in the stiffness matrix is also proposed and analyzed. Finally, numerical results are presented to illustrate the theoretical analysis.
LA - eng
KW - Finite elements; interior penalty; stabilization methods; Friedrichs' systems; first-order PDE's.; finite elements; first-order PDEs; numerical results; convergence
UR - http://eudml.org/doc/250077
ER -

References

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  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 a posteriori estimates for nonconforming mixed finite element methods. ESAIM: M2AN38 (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: M2AN40 (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.  

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