Previous |  Up |  Next


nonlinear convection-diffusion problem; barycentric finite volumes; Crouzeix-Raviart nonconforming piecewise linear finite elements; monotone finite volume scheme; discrete maximum principle; a priori estimates; convergence of the method
We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.
[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), 1–26. DOI 10.1007/BF02238077 | MR 1314953
[2] D. Adam and H.-G. Roos: A nonconforming exponentially field fitted finite element method I: The interpolation error. Preprint MATH-NM-06-1993, Technische Universität Dresden, 1993. MR 1230386
[3] P. Arminjon, A. Dervieux, L. Fezoui, H. Steve and B. Stoufflet: Non-oscillatory schemes for multidimensional Euler calculations with unstructured grids. In Notes on Numerical Fluid Mechanics Volume 24 (Nonlinear Hyperbolic Equations—Theory, Computation Methods and Applications), Ballman J. and Jeltsch R. (eds.), Technical report, Vieweg, Braunschweig-Wiesbaden, 1989 1989, pp. 1–10. MR 0991347
[4] P. G. Ciarlet: The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam, 1979. MR 0520174
[5] M. Crouzeix and P.-A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO, Anal. Numér. 7 (1973), 733–76. MR 0343661
[6] V. Dolejší and P. Angot: Finite Volume Methods on Unstructured Meshes for Compressible Flows. In Finite Volumes for Complex Applications (Problems and Perspectives), F. Benkhaldoun and R. Vilsmeier (eds.), ISBN 2-86601-556-8, Hermes, Rouen, 1996, pp. 667–674.
[7] V. Dolejší, M. Feistauer and J. Felcman: On the discrete Friedrichs inequality for nonconforming finite elements. Preprint, Faculty of Mathematics and Physics, Charles University, Prague 1998. MR 1704954
[8] M. Feistauer: Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Monographs and Surveys in Pure and Applied Mathematics 67, Harlow, 1993. Zbl 0819.76001
[9] M. Feistauer and J. Felcman: Convection-diffusion problems and compressible Navier-Stokes equations. In The Mathematics of Finite Elements and Applications, J. R. Whiteman (ed.), Wiley, 1996, pp. 175–194.
[10] M. Feistauer, J. Felcman and V. Dolejší: Numerical simulation of compresssible viscous flow through cascades of profiles. ZAMM 76 (1996), 297–300.
[11] 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
[12] 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
[13] 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. (to appear). MR 1706727
[14] M. Feistauer, J. Felcman, M. Rokyta and Z. Vlášek: Finite element solution of flow problems with trailing conditions. J. Comput. Appl. Math. 44 (1992), 131–145. DOI 10.1016/0377-0427(92)90008-L | MR 1197680
[15] M. Feistauer, J. Slavík and P. Stupka: On the convergence of the combined finite volume—finite element method for nonlinear convection-diffusion problems. Explicit schemes. Numer. Methods Partial Differential Equations (submitted). MR 1674294
[16] J. Felcman: Finite volume solution of the inviscid compressible fluid flow. ZAMM 72 (1992), 513–516. Zbl 0825.76666
[17] J. Felcman and V. Dolejší: Adaptive methods for the solution of the euler equations in elements of the blade bachines. ZAMM 76 (1996), 301–304.
[18] 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 and Sons, Stuttgart, 1994, pp. 894–901.
[19] J. Felcman and G. Warnecke: Adaptive computational methods for gas flow. In Proceedings of the Prague Mathematical Conference, Prague, ICARIS, 1996, pp. 99–104. MR 1703464
[20] 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.
[21] R. Glowinski, J. L. Lions and R. Trémolières: Analyse numérique des inéquations variationnelles. Dunod, Paris, 1976.
[22] 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
[23] C. Johnson: Finite element methods for convection-diffusion problems. In Computing Methods in Engineering and Applied Sciences V, Glowinski R. and Lions J. L. (eds.), North-Holland, Amsterdam, 1981. MR 0784648
[24] D. Kröner: Numerical Schemes for Conservation Laws. Wiley-Teubner, Stuttgart, 1997. MR 1437144
[25] A. Kufner, O. John and S. Fučík: Function Spaces. Academia, Prague, 1977. MR 0482102
[26] K. W. Morton: Numerical Solution of Convection-Diffusion Problems. Chapman & Hall, London, 1996. MR 1445295 | Zbl 0861.65070
[27] K. Ohmori and 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
[28] H.-G. Roos, M. Stynes and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, 24, Springer-Verlag, Berlin, 1996. MR 1477665
[29] F. Schieweck and L. Tobiska: A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation. $M^2AN$ 23 (1989), 627–647. MR 1025076
[30] 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
[31] R. Temam: Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford, 1979. MR 0603444 | Zbl 0454.35073
[32] L. Tobiska: Full and weighted upwind finite element methods. In Splines in Numerical Analysis Mathematical Research Volume, Volume 32, J. W. Schmidt, H. Spath (eds.), Akademie-Verlag, Berlin, 1989. MR 1004263 | Zbl 0685.65074
[33] G. Vijayasundaram: Transonic flow simulation using an upstream centered scheme of Godunov in finite elements. J. Comp. Phys. 63 (1986), 416–433. DOI 10.1016/0021-9991(86)90202-0 | MR 0835825
[34] G. Zhou: A local ${L}^2$-error analysis of the streamline diffusion method for nonstationary convection-diffusion systems. $M^2AN $ 29 (1995), 577–603. MR 1352863
[35] G. Zhou and 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