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 convergence rates in the -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.
Mots-clés : finite elements, interior penalty, stabilization methods, Friedrichs' systems, first-order PDE's
@article{M2AN_2007__41_1_55_0,
author = {Burman, Erik and Ern, Alexandre},
title = {A continuous finite element method with face penalty to approximate {Friedrichs'} systems},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {55--76},
publisher = {EDP-Sciences},
volume = {41},
number = {1},
year = {2007},
doi = {10.1051/m2an:2007007},
zbl = {1129.65083},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007007/}
}
TY - JOUR AU - Burman, Erik AU - Ern, Alexandre TI - A continuous finite element method with face penalty to approximate Friedrichs' systems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 55 EP - 76 VL - 41 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007007/ DO - 10.1051/m2an:2007007 LA - en ID - M2AN_2007__41_1_55_0 ER -
%0 Journal Article %A Burman, Erik %A Ern, Alexandre %T A continuous finite element method with face penalty to approximate Friedrichs' systems %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 55-76 %V 41 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007007/ %R 10.1051/m2an:2007007 %G en %F M2AN_2007__41_1_55_0
Burman, Erik; Ern, Alexandre. A continuous finite element method with face penalty to approximate Friedrichs' systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 1, pp. 55-76. doi : 10.1051/m2an:2007007. http://www.numdam.org/articles/10.1051/m2an:2007007/
[1] , The finite element method with penalty. Math. Comp. 27 (1973) 221-228. | Zbl
[2] and, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863-875. | Zbl
[3] , Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31 (1977) 45-59. | Zbl
[4] and, 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). | MR | Zbl
[5] , A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43 (2005) 2012-2033. | Zbl
[6] and, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437-1453. | Zbl
[7] and, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410. | Zbl
[8] and, Discontinuous and continuous finite element methods with interior penalty for hyperbolic problems. J. Numer. Math (2005) Submitted (EPFL-IACS report 17.2005).
[9] ,, and. First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem. SIAM J. Numer. Anal. 35 (1998) 320-335. | Zbl
[10] , and, Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lect. Notes Phys. 58, Springer-Verlag, Berlin (1976). | MR
[11] and, Residual and hierarchical a posteriori estimates for nonconforming mixed finite element methods. ESAIM: M2AN 38 (2004) 903-929. | Numdam | Zbl
[12] and, Theory and Practice of Finite Elements. Appl. Math. Sci. 159, Springer-Verlag, New York, NY (2004). | MR | Zbl
[13] and, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753-778. | Zbl
[14] and, Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order PDEs. SIAM J. Numer. Anal. 44 (2006) 2363-2388. | Zbl
[15] and, Discontinuous Galerkin methods for Friedrichs' systems. III. Multi-field theories with partial coercivity. SIAM J. Numer. Anal. (2006) Submitted (CERMICS report 2006-320). | Zbl
[16] and, Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM: M2AN 40 (2006) 29-48. | Numdam
[17] and, Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1999) 935-952. | Zbl
[18] , Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958) 333-418. | Zbl
[19] and, FreeFEM++ Manual. Laboratoire Jacques-Louis Lions, University Paris VI (2005).
[20] and, Element-oriented and edge-oriented local error estimators for non-conforming finite element methods. RAIRO Math. Model. Anal. Numer. 30 (1996) 237-263. | Numdam | Zbl
[21] , Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions. Ph.D. thesis, University of Oxford (2004).
[22] and, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1-26. | Zbl
[23] and, A-posteriori error estimates for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374-2399. | Zbl
[24] , Finite element methods for symmetric hyperbolic equations. Numer. Math. 21 (1973/74) 244-255. | Zbl
[25] , 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] and. 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. | Zbl
Cité par Sources :
