Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

Philippe Angot; Vít Dolejší; Miloslav Feistauer; Jiří Felcman

Applications of Mathematics (1998)

  • Volume: 43, Issue: 4, page 263-310
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Angot, Philippe, et al. "Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems." Applications of Mathematics 43.4 (1998): 263-310. <http://eudml.org/doc/33012>.

@article{Angot1998,
abstract = {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.},
author = {Angot, Philippe, Dolejší, Vít, Feistauer, Miloslav, Felcman, Jiří},
journal = {Applications of Mathematics},
keywords = {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; finite volume method; implicit time discretization; nonlinear convection-diffusion problem; two-dimensional polygonal domain; nonconforming triangular piecewise-linear finite element method; discrete maximum principle; stability; a priori error estimates; convergence theorem},
language = {eng},
number = {4},
pages = {263-310},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems},
url = {http://eudml.org/doc/33012},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Angot, Philippe
AU - Dolejší, Vít
AU - Feistauer, Miloslav
AU - Felcman, Jiří
TI - Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems
JO - Applications of Mathematics
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 43
IS - 4
SP - 263
EP - 310
AB - 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.
LA - eng
KW - 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; finite volume method; implicit time discretization; nonlinear convection-diffusion problem; two-dimensional polygonal domain; nonconforming triangular piecewise-linear finite element method; discrete maximum principle; stability; a priori error estimates; convergence theorem
UR - http://eudml.org/doc/33012
ER -

References

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  1. 10.1007/BF02238077, Computing 54 (1995), 1–26. (1995) MR1314953DOI10.1007/BF02238077
  2. A nonconforming exponentially field fitted finite element method I: The interpolation error, Preprint MATH-NM-06-1993, Technische Universität Dresden, 1993. (1993) MR1230386
  3. 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. (1989 1989) MR0991347
  4. The Finite Elements Method for Elliptic Problems, North-Holland, Amsterdam, 1979. (1979) MR0520174
  5. Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO, Anal. Numér. 7 (1973), 733–76. (1973) MR0343661
  6. 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. (1996) 
  7. On the discrete Friedrichs inequality for nonconforming finite elements, Preprint, Faculty of Mathematics and Physics, Charles University, Prague 1998. MR1704954
  8. Mathematical Methods in Fluid Dynamics, Longman Scientific & Technical, Monographs and Surveys in Pure and Applied Mathematics 67, Harlow, 1993. (1993) Zbl0819.76001
  9. 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. (1996) 
  10. Numerical simulation of compresssible viscous flow through cascades of profiles, ZAMM 76 (1996), 297–300. (1996) 
  11. 10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N, Numer. Methods Partial Differential Equations 13 (1997), 163–190. (1997) MR1436613DOI10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N
  12. 10.1016/0377-0427(95)00051-8, J. Comput. Appl. Math. 63 (1995), 179–199. (1995) MR1365559DOI10.1016/0377-0427(95)00051-8
  13. Error estimates of a combined finite volume—finite element method for nonlinear convection-diffusion problems, SIAM J. Numer. Anal. (to appear). (to appear) MR1706727
  14. 10.1016/0377-0427(92)90008-L, J. Comput. Appl. Math. 44 (1992), 131–145. (1992) MR1197680DOI10.1016/0377-0427(92)90008-L
  15. On the convergence of the combined finite volume—finite element method for nonlinear convection-diffusion problems. Explicit schemes, Numer. Methods Partial Differential Equations (submitted). MR1674294
  16. Finite volume solution of the inviscid compressible fluid flow, ZAMM 72 (1992), 513–516. (1992) Zbl0825.76666
  17. Adaptive methods for the solution of the euler equations in elements of the blade bachines, ZAMM 76 (1996), 301–304. (1996) 
  18. 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. (1994) 
  19. Adaptive computational methods for gas flow, In Proceedings of the Prague Mathematical Conference, Prague, ICARIS, 1996, pp. 99–104. (1996) MR1703464
  20. 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. (1995) 
  21. Analyse numérique des inéquations variationnelles, Dunod, Paris, 1976. (1976) 
  22. 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. (1983) Zbl0508.65049MR0683102
  23. 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. (1981) MR0784648
  24. Numerical Schemes for Conservation Laws, Wiley-Teubner, Stuttgart, 1997. (1997) MR1437144
  25. Function Spaces, Academia, Prague, 1977. (1977) MR0482102
  26. Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, 1996. (1996) Zbl0861.65070MR1445295
  27. 10.1051/m2an/1984180303091, RAIRO, Anal. Numér. 18 (1984), 309–322. (1984) MR0751761DOI10.1051/m2an/1984180303091
  28. Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, 24, Springer-Verlag, Berlin, 1996. (1996) MR1477665
  29. A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation, 23 (1989), 627–647. (1989) MR1025076
  30. 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. (1972) Zbl0264.65068MR0413554
  31. Navier-Stokes Equations, North-Holland, Amsterdam-New York-Oxford, 1979. (1979) Zbl0454.35073MR0603444
  32. 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. (1989) Zbl0685.65074MR1004263
  33. 10.1016/0021-9991(86)90202-0, J. Comp. Phys. 63 (1986), 416–433. (1986) MR0835825DOI10.1016/0021-9991(86)90202-0
  34. A local L 2 -error analysis of the streamline diffusion method for nonstationary convection-diffusion systems, 29 (1995), 577–603. (1995) MR1352863
  35. 10.1002/(SICI)1098-2426(199601)12:1<123::AID-NUM7>3.0.CO;2-U, Numer. Methods Partial Differential Equations 12 (1996), 123–145. (1996) MR1363866DOI10.1002/(SICI)1098-2426(199601)12:1<123::AID-NUM7>3.0.CO;2-U

Citations in EuDML Documents

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  1. Stéphane Flotron, Jacques Rappaz, Conservation schemes for convection-diffusion equations with Robin boundary conditions
  2. Vít Dolejší, Miloslav Feistauer, Christoph Schwab, On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow
  3. Vít Dolejší, Miloslav Feistauer, Jiří Felcman, Výpočtová matematika a počítačová dynamika tekutin
  4. Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations
  5. Vít Dolejší, Miloslav Feistauer, Jiří Felcman, Alice Kliková, Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems
  6. Mario Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations
  7. Mario Ohlberger, error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

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