OPTIMAL AND PRESSURE-INDEPENDENT L2 VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

doi:10.4208/jcm.1411-m4499

Journal of Computational Mathematics ›› 2015, Vol. 33 ›› Issue (2): 191-208.DOI: 10.4208/jcm.1411-m4499

Previous Articles Next Articles

OPTIMAL AND PRESSURE-INDEPENDENT L² VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

C. Brennecke¹, A. Linke², C. Merdon², J. Schöberl³

1. Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistr. 101, 8092 Zürich, Switzerland;
2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany;
3. TU Wien, Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8-10/101,1040 Wien, Austria

Received:2014-03-03 Revised:2014-11-17 Online:2015-03-15 Published:2015-03-13
Supported by:
The research documented in this paper has been partially funded in the framework of the project “Macroscopic Modeling of Transport and Reaction Processes in Magnesium-Air-Batteries” (Grant 03EK3027D) under the research initiative “Energy storage” of the German Federal government.

PDF ( 186 ) Cited

C. Brennecke, A. Linke, C. Merdon, J. Schöberl. OPTIMAL AND PRESSURE-INDEPENDENT L² VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS[J]. Journal of Computational Mathematics, 2015, 33(2): 191-208.

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H¹ velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressureindependent L² velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

Key words: Variational crime, Crouzeix-Raviart finite element, Divergence-free mixed method, Incompressible Navier-Stokes equations, A priori error estimates

CLC Number:

[1] G. Acosta and R.G. Durán, The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations, SIAM J. Numer. Anal., 37:1 (1999), 18-36 (electronic).

[2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag New York, Inc., New York, NY, USA, 1991.

[3] E. Burman and A. Linke, Stabilized finite element schemes for incompressible flow using Scott- Vogelius elements, Appl. Numer. Math., 58:11 (2008), 1704-1719.

[4] C. Carstensen, J. Gedicke and D. Rim, Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Thomas finite element methods, J. Comput. Math., 30:4 (2012), 337-353.

[5] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7:R-3 (1973), 33-75.

[6] O. Dorok, W. Grambow and L. Tobiska, Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier-Stokes Equations, Notes on Numerical Fluid Mechanics: Numerical Methods for the Navier-Stokes Equations. Proceedings of the International Workshop held at Heidelberg, October 1993, ed. by F.-K. Hebeker, R. Rannacher and G. Wittum 47 (1994), 50-61.

[7] L. Franca and T. Hughes, Two classes of mixed finite element methods, Computer Methods in Applied Mechanics and Engineering, 69:1 (1988), 89-129.

[8] J.-F. Gerbeau, C. Le Bris and M. Bercovier, Spurious velocities in the steady flow of an incompressible fluid subjected to external forces, International Journal for Numerical Methods in Fluids, 25:6 (1997), 679-695 (Anglais).

[9] K.J. Galvin, A. Linke, L.G. Rebholz and N.E. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg., 237/240 (2012), 166-176.

[10] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986.

[11] E. Jenkins, V. John, L. Rebholz and A. Linke, On the parameter choice in grad-div stabilization for the Stokes equations, Advances in Computational Mathematics (2014), in press.

[12] A. Linke, Divergence-free mixed finite elements for the incompressible Navier-Stokes equation, Ph.D. thesis, University of Erlangen, 2008.

[13] A. Linke, Collision in a cross-shaped domain - a steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD, Computer Methods in Applied Mechanics and Engineering, 198:41-44 (2009), 3278-3286.

[14] A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., 268 (2014), 782-800.

[15] M.A. Olshanskii, G. Lube, T. Heister and J. Löwe, Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198:49-52 (2009), 3975-3988.

[16] M. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations, Math. Comp., 73:248 (2004), 1699-1718.

[17] P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Springer, Berlin, 1977, pp. 292-315. Lecture Notes in Math., Vol. 606.

[18] R. Temam, Navier-Stokes Equations, Elsevier, North-Holland, 1991.

[1]	Michael Holst, Christopher Tiee. FINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES [J]. Journal of Computational Mathematics, 2018, 36(6): 792-832.
[2]	Liang Ge, Tongjun Sun. A SPARSE GRID STOCHASTIC COLLOCATION AND FINITE VOLUME ELEMENT METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY RANDOM ELLIPTIC EQUATIONS [J]. Journal of Computational Mathematics, 2018, 36(2): 310-330.
[3]	Hongfei Fu, Hongxing Rui. A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2015, 33(2): 113-127.
[4]	Tianliang Hou, Yanping Chen. MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2015, 33(2): 158-178.
[5]	Hongfei Fu, Hongxing Rui, Hui Guo. A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2013, 31(1): 88-106.
[6]	Lan Chieh HUANG. NUMERICAL SOLUTION OF THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON THE CURVILINEAR HALF-STAGGERED MESH NUMERICAL SOLUTION OF THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON THE CURVILINEAR HALF-STAGGERED MESH [J]. Journal of Computational Mathematics, 2000, 18(5): 521-540.

Viewed

Full text

Abstract

OPTIMAL AND PRESSURE-INDEPENDENT L2 VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

扫码分享

OPTIMAL AND PRESSURE-INDEPENDENT L² VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS