Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-30T20:22:43.506Z Has data issue: false hasContentIssue false
  • English
  • Français

A continuous finite element method with face penaltyto approximate Friedrichs' systems

Published online by Cambridge University Press:  26 April 2007

Erik Burman  and
Alexandre Ern
Erik Burman
Affiliation:
Department of Mathematics, École Polytechnique Fédérale de Lausanne, Switzerland. erik.burman@epfl.ch
Alexandre Ern
Affiliation:
CERMICS, École des ponts, ParisTech, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France. ern@cermics.enpc.fr
Get access

Abstract

A continuous finite element method to approximate Friedrichs' systems isproposed and analyzed. Stability is achieved by penalizing the jumpsacross meshinterfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence ratesin the graph norm and suboptimal of order ½ convergence rates inthe L 2-norm. A variant of the method specialized toFriedrichs' systems associated with elliptic PDE's in mixed form andreducing the number of nonzero entries in the stiffness matrix is alsoproposed and analyzed. Finally, numerical results are presented to illustrate thetheoretical analysis.

Keywords

Finite elementsinterior penaltystabilization methodsFriedrichs' systemsfirst-order PDE's.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 55 - 76
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Babuška, I., The finite element method with penalty. Math. Comp. 27 (1973) 221–228. CrossRef
Babuška, I. and Zlámal, M., Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863–875. CrossRef
Baker, G.A., Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31 (1977) 45–59. CrossRef
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).
Burman, E., A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43 (2005) 2012–2033. CrossRef
Burman, E. and Hansbo, P., Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437–1453. CrossRef
Burman, E. and Hansbo, P., Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393–2410. CrossRef
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).
Cai, Z., Manteuffel, T.A., McCormick, S.F. and Parter, S.V.. First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem. SIAM J. Numer. Anal. 35 (1998) 320–335. CrossRef
J. Douglas, Jr., and T. Dupont, Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lect. Notes Phys. 58, Springer-Verlag, Berlin (1976).
El Alaoui, L. and Ern, A., Residual and hierarchical a posteriori estimates for nonconforming mixed finite element methods. ESAIM: M2AN 38 (2004) 903–929. CrossRef
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Appl. Math. Sci. 159, Springer-Verlag, New York, NY (2004).
Ern, A. and Guermond, J.-L., Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753–778. CrossRef
Ern, A. and Guermond, J.-L., Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order PDEs. SIAM J. Numer. Anal. 44 (2006) 2363–2388. CrossRef
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).
Ern, A. and Guermond, J.-L., Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM: M2AN 40 (2006) 29–48. CrossRef
Falk, R.S. and Richter, G.R., Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1999) 935–952. CrossRef
Friedrichs, K.O., Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958) 333–418. CrossRef
F. Hecht and O. Pironneau, FreeFEM++ Manual. Laboratoire Jacques-Louis Lions, University Paris VI (2005).
Hoppe, R.H.W. and Wohlmuth, B., Element-oriented and edge-oriented local error estimators for non-conforming finite element methods. RAIRO Math. Model. Anal. Numer. 30 (1996) 237–263. CrossRef
M. Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions. Ph.D. thesis, University of Oxford (2004).
Johnson, C. and Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1–26. CrossRef
Karakashian, O. and Pascal, F., A-posteriori error estimates for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. CrossRef
P. Lesaint, Finite element methods for symmetric hyperbolic equations. Numer. Math. 21 (1973/74) 244–255.
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).
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.