Previous |  Up |  Next


nonlinear convection-diffusion problem; compressible Navier-Stokes equations; cascade flow; barycentric finite volumes; Crouzeix-Raviart nonconforming piecewise linear finite elements; monotone finite volume scheme; discrete maximum principle; a priori estimates; error estimates
The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the $L^2(L^2)$ and $L^2(H^1)$ error estimates are established. At the end of the paper, some computational results are presented demonstrating the application of the method to the solution of viscous gas flow.
[1] D. Adam, A. Felgenhauer, H.-G. Roos and M. Stynes: A nonconforming finite element method for a singularly perturbed boundary value problem. Computing 54 (1995), 1–25. DOI 10.1007/BF02238077 | MR 1314953
[2] Ph. Angot, V. Dolejší, M. Feistauer and J. Felcman: Analysis of a combined barycentric finite volume-nonconforming finite element method for nonlinear convection-diffusion problems. Appl. Math. 43 (1998), 263–310. DOI 10.1023/A:1023217905340 | MR 1627989
[3] P. Arminjon, A. Madrane: A mixed finite volume/finite element method for 2-dimensional compressible Navier-Stokes equations on unstructured grids. In: Hyperbolic Problems: Theory, Numerics, Applications, M. Fey, R. Jeltsch (eds.), Birkhäuser, Basel, 1999. MR 1715728
[4] P. G.  Ciarlet: The Finite Elements Method for Elliptic Problems. Studies in Mathematics and its Applications, Vol. 4. North-Holland, Amsterdam, 1978. MR 0520174
[5] M. Crouzeix, P.-A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7 (1973), 33–75. MR 0343661
[6] V. Dolejší: Sur des méthodes combinant des volumes finis et des éléments finis pour le calcul d’écoulements compressibles sur des maillages non structurés. PhD Dissertation, Charles University Prague—L’Université Méditerranée Marseille, 1998.
[7] V. Dolejší: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1 (1998), 165–178. DOI 10.1007/s007910050015
[8] V. Dolejší, P. Angot: Finite volume methods on unstructured meshes for compressible flows. In: Finite Volumes for Complex Applications (Problems and Perspectives),, F. Benkhaldoun, R. Vilsmeier (eds.), Hermes, Rouen, 1996, pp. 667–674.
[9] R. Eymand, T. Gallouët and R. Herbin: Finite Volume Methods. Technical Report 97-19, Centre de Mathématiques et d’Informatique. Université de Provence, Marseille, 1997.
[10] V. Dolejší, M. Feistauer and J. Felcman: On the discrete Friedrichs inequality for nonconforming finite elements. Numer. Funct. Anal. Optim. 20 (1999), 437–447. DOI 10.1080/01630569908816904 | MR 1704954
[11] M. Feistauer: Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 67. Longman Scientific & Technical, Harlow, 1993. MR 1266627
[12] M. Feistauer, V. Dolejší, J. Felcman and A. Kliková: Adaptive mesh refinement for problems of fluid dynamics. In: Proc. of Colloquium Fluid Dynamics ’99, P. Jonáš, V.  Uruba (eds.), Institute of Thermomechanics, Academy of Sciences, Prague, 1999, pp. 53–60.
[13] M. Feistauer, J. Felcman: Convection-diffusion problems and compressible Navier-Stokes equations. In: The Mathematics of Finite Elements and Applications, J. R. Whiteman (ed.), John Wiley & Sons, 1997, pp. 175–194.
[14] M. Feistauer, J. Felcman and V. Dolejší: Numerical simulation of compresssible viscous flow through cascades of profiles. Z. Angew. Math. Mech. 76 (1996), 297–300.
[15] M. Feistauer, J. Felcman and M. Lukáčová: Combined finite elements-finite volume solution of compressible flow. J. Comput. Appl. Math. 63 (1995), 179–199. DOI 10.1016/0377-0427(95)00051-8 | MR 1365559
[16] M. Feistauer, J. Felcman and M. Lukáčová: On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Numer. Methods Partial Differential Equations 13 (1997), 163–190. DOI 10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N | MR 1436613
[17] M. Feistauer, J. Felcman, M. Lukáčová and G. Warnecke: Error estimates of a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal. 36 (1999), 1528–1548. DOI 10.1137/S0036142997314695 | MR 1706727
[18] M. Feistauer, J. Slavík and P. Stupka: On the convergence of the combined finite volume-finite element method for nonlinear convection-diffusion problems. II. Explicit schemes. Numer. Methods Partial Differential Equations 15 (1999), 215–235. DOI 10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1 | MR 1674294
[19] J. Felcman: Finite volume solution of the inviscid compressible fluid flow. Z. Angew. Math. Mech. 72 (1992), 513–516. Zbl 0825.76666
[20] J. Felcman, V. Dolejší: Adaptive methods for the solution of the Euler equations in elements of the blade machines. Z. Angew. Math. Mech. 76 (1996), 301–304.
[21] J. Felcman, V. Dolejší and M. Feistauer: Adaptive finite volume method for the numerical solution of the compressible Euler equations. In: Computational Fluid Dynamics ’94, J. Périaux, S. Wagner, E. H. Hirschel and R. Piva (eds.), John Wiley & Sons, Stuttgart, 1994, pp. 894–901.
[22] J. Felcman, G. Warnecke: Adaptive computational methods for gas flow. In: Proceedings of the Prague Mathematical Conference, K. Segeth (ed.), ICARIS, Prague, 1996, pp. 99–104. MR 1703464
[23] J. Fořt, M. Huněk, K. Kozel and M. Vavřincová: Numerical simulation of steady and unsteady flows through plane cascades. In: Numerical Modeling in Continuum Mechanics II, R. Ranacher, M. Feistauer and K. Kozel (eds.), Faculty of Mathematics and Physics, Charles Univ., Prague, 1995, pp. 95–102.
[24] H. Gajewski, K. Gröger and K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974. MR 0636412
[25] T. Ikeda: Maximum principle in finite element models for convection-diffusion phenomena. In: Mathematics Studies 76, Lecture Notes in Numerical and Applied Analysis Vol. 4, North-Holland, Amsterdam-New York-Oxford, 1983. MR 0683102 | Zbl 0508.65049
[26] C. Johnson: Finite element methods for convection-diffusion problems. In: Computing Methods in Engineering and Applied Sciences V., R. Glowinski, J. L. Lions (eds.), North-Holland, Amsterdam, 1981. MR 0784648
[27] C. Johnson: Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge, 1988.
[28] A. Kliková: Finite Volume—Finite Element Solution of Compressible Flow. Doctoral Thesis. Charles University Prague, 2000.
[29] A. Kliková, M. Feistauer and J. Felcman: Adaptive methods for problems of fluid dynamics. In: Software and Algorithms of Numerical Mathematics ’99, J. Holenda, I. Marek (eds.), West-Bohemian University, Pilsen, 1999.
[30] D. Kröner: Numerical Schemes for Conservation Laws. Wiley & Teuner, Chichester, 1997. MR 1437144
[31] D. Kröner, M. Rokyta: Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31 (1994), 324–343. DOI 10.1137/0731017 | MR 1276703
[32] M. Křížek, Qun Lin: On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 3 (1993), 59–69.
[33] M. Křížek, P.  Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 50. Longman Scientific & Technical, Harlow, 1990. MR 1066462
[34] A. Kufner, O. John and S. Fučík: Function Spaces. Academia, Prague, 1977. MR 0482102
[35] J. M. Melenk and C. Schwab: The $hp$ streamline diffusion finite element method for convection dominated problems in one space dimension. East-West J. Numer. Math. 7 (1999), 31–60. MR 1683935
[36] K. W. Morton: Numerical Solution of Convection-Diffusion Problems. Chapman & Hall, London, 1996. MR 1445295 | Zbl 0861.65070
[37] K. Ohmori, T. Ushijima: A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations. RAIRO Anal. Numér. 18 (1984), 309–322. MR 0751761
[38] S. Osher, F. Solomon: Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comp. 38 (1982), 339–374. DOI 10.1090/S0025-5718-1982-0645656-0 | MR 0645656
[39] H.-G. Roos, M. Stynes and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, Vol. 24. Springer-Verlag, Berlin, 1996. MR 1477665
[40] F. Schieweck, L. Tobiska: A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation. RAIRO Modél. Math. Anal. Numér. 23 (1989), 627–647. MR 1025076
[41] C. Schwab: $p$- and $hp$- Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Clarendon Press, Oxford, 1998. MR 1695813 | Zbl 0910.73003
[42] S. P. Spekreijse: Multigrid Solution of the Steady Euler Equations. Centrum voor Wiskunde en Informatica, Amsterdam, 1987. MR 0942891
[43] G. Strang: Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method, A. K. Aziz (ed.), Academic Press, New York, 1972, pp. 689–710. MR 0413554 | Zbl 0264.65068
[44] M. Šťastný, P. Šafařík: Experimental analysis data on the transonic flow past a plane turbine cascade. ASME Paper 90-GT-313, New York, 1990.
[45] R. Temam: Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford, 1979. MR 0603444 | Zbl 0454.35073
[46] L. Tobiska: Full and weighted upwind finite element methods. In: Splines in Numerical Analysis Mathematical Research Volume, Vol. 32, J. W. Schmidt, H. Spath (eds.), Akademie-Verlag, Berlin, 1989. MR 1004263 | Zbl 0685.65074
[47] G. Vijayasundaram: Transonic flow simulation using an upstream centered scheme of Godunov in finite elements. J. Comp. Phys. 63 (1986), 416–433. MR 0835825
[48] G. Zhou: A local ${L}^2$-error analysis of the streamline diffusion method for nonstationary convection-diffusion systems. RAIRO Modél. Math. Anal. Numér. 29 (1995), 577–603. MR 1352863
[49] G. Zhou, R. Rannacher: Pointwise superconvergence of streamline diffusion finite-element method. Numer. Methods Partial Differential Equations 12 (1996), 123–145. DOI 10.1002/(SICI)1098-2426(199601)12:1<123::AID-NUM7>3.0.CO;2-U | MR 1363866
Partner of
EuDML logo