Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems

Vít Dolejší; Miloslav Feistauer; Jiří Felcman; Alice Kliková

Applications of Mathematics (2002)

  • Volume: 47, Issue: 4, page 301-340
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Dolejší, Vít, et al. "Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems." Applications of Mathematics 47.4 (2002): 301-340. <http://eudml.org/doc/33118>.

@article{Dolejší2002,
abstract = {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.},
author = {Dolejší, Vít, Feistauer, Miloslav, Felcman, Jiří, Kliková, Alice},
journal = {Applications of Mathematics},
keywords = {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; nonlinear convection-diffusion problem; compressible Navier-Stokes equations; cascade flow; barycentric finite volumes; discrete maximum principle; a priori estimates; error estimates},
language = {eng},
number = {4},
pages = {301-340},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems},
url = {http://eudml.org/doc/33118},
volume = {47},
year = {2002},
}

TY - JOUR
AU - Dolejší, Vít
AU - Feistauer, Miloslav
AU - Felcman, Jiří
AU - Kliková, Alice
TI - Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems
JO - Applications of Mathematics
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 4
SP - 301
EP - 340
AB - 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.
LA - eng
KW - 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; nonlinear convection-diffusion problem; compressible Navier-Stokes equations; cascade flow; barycentric finite volumes; discrete maximum principle; a priori estimates; error estimates
UR - http://eudml.org/doc/33118
ER -

References

top
  1. 10.1007/BF02238077, Computing 54 (1995), 1–25. (1995) MR1314953DOI10.1007/BF02238077
  2. 10.1023/A:1023217905340, Appl. Math. 43 (1998), 263–310. (1998) MR1627989DOI10.1023/A:1023217905340
  3. 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. (1999) MR1715728
  4. The Finite Elements Method for Elliptic Problems. Studies in Mathematics and its Applications, Vol. 4, North-Holland, Amsterdam, 1978. (1978) MR0520174
  5. Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numér. 7 (1973), 33–75. (1973) MR0343661
  6. 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. 10.1007/s007910050015, Comput. Vis. Sci. 1 (1998), 165–178. (1998) DOI10.1007/s007910050015
  8. 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. (1996) 
  9. Finite Volume Methods. Technical Report 97-19, Centre de Mathématiques et d’Informatique, Université de Provence, Marseille, 1997. (1997) 
  10. 10.1080/01630569908816904, Numer. Funct. Anal. Optim. 20 (1999), 437–447. (1999) MR1704954DOI10.1080/01630569908816904
  11. Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 67, Longman Scientific & Technical, Harlow, 1993. (1993) MR1266627
  12. 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. (1999) 
  13. 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. (1997) 
  14. Numerical simulation of compresssible viscous flow through cascades of profiles, Z. Angew. Math. Mech. 76 (1996), 297–300. (1996) 
  15. 10.1016/0377-0427(95)00051-8, J. Comput. Appl. Math. 63 (1995), 179–199. (1995) MR1365559DOI10.1016/0377-0427(95)00051-8
  16. 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
  17. 10.1137/S0036142997314695, SIAM J. Numer. Anal. 36 (1999), 1528–1548. (1999) MR1706727DOI10.1137/S0036142997314695
  18. 10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1, Numer. Methods Partial Differential Equations 15 (1999), 215–235. (1999) MR1674294DOI10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1
  19. Finite volume solution of the inviscid compressible fluid flow, Z. Angew. Math. Mech. 72 (1992), 513–516. (1992) Zbl0825.76666
  20. Adaptive methods for the solution of the Euler equations in elements of the blade machines, Z. Angew. Math. Mech. 76 (1996), 301–304. (1996) 
  21. 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. (1994) 
  22. Adaptive computational methods for gas flow, In: Proceedings of the Prague Mathematical Conference, K. Segeth (ed.), ICARIS, Prague, 1996, pp. 99–104. (1996) MR1703464
  23. 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) 
  24. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. (1974) MR0636412
  25. 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
  26. 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. (1981) MR0784648
  27. Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, 1988. (1988) 
  28. Finite Volume—Finite Element Solution of Compressible Flow. Doctoral Thesis, Charles University Prague, 2000. (2000) 
  29. 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. (1999) 
  30. Numerical Schemes for Conservation Laws, Wiley & Teuner, Chichester, 1997. (1997) MR1437144
  31. 10.1137/0731017, SIAM J. Numer. Anal. 31 (1994), 324–343. (1994) MR1276703DOI10.1137/0731017
  32. On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math. 3 (1993), 59–69. (1993) 
  33. Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 50, Longman Scientific & Technical, Harlow, 1990. (1990) MR1066462
  34. Function Spaces, Academia, Prague, 1977. (1977) MR0482102
  35. The h p streamline diffusion finite element method for convection dominated problems in one space dimension, East-West J. Numer. Math. 7 (1999), 31–60. (1999) MR1683935
  36. Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, 1996. (1996) Zbl0861.65070MR1445295
  37. 10.1051/m2an/1984180303091, RAIRO Anal. Numér. 18 (1984), 309–322. (1984) MR0751761DOI10.1051/m2an/1984180303091
  38. 10.1090/S0025-5718-1982-0645656-0, Math. Comp. 38 (1982), 339–374. (1982) MR0645656DOI10.1090/S0025-5718-1982-0645656-0
  39. Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, Vol. 24, Springer-Verlag, Berlin, 1996. (1996) MR1477665
  40. 10.1051/m2an/1989230406271, RAIRO Modél. Math. Anal. Numér. 23 (1989), 627–647. (1989) MR1025076DOI10.1051/m2an/1989230406271
  41. p - and h p - Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics, Clarendon Press, Oxford, 1998. (1998) Zbl0910.73003MR1695813
  42. Multigrid Solution of the Steady Euler Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. (1987) MR0942891
  43. 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
  44. Experimental analysis data on the transonic flow past a plane turbine cascade, ASME Paper 90-GT-313, New York, 1990. (1990) 
  45. Navier-Stokes Equations, North-Holland, Amsterdam-New York-Oxford, 1979. (1979) Zbl0454.35073MR0603444
  46. 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. (1989) Zbl0685.65074MR1004263
  47. Transonic flow simulation using an upstream centered scheme of Godunov in finite elements, J. Comp. Phys. 63 (1986), 416–433. (1986) MR0835825
  48. 10.1051/m2an/1995290505771, RAIRO Modél. Math. Anal. Numér. 29 (1995), 577–603. (1995) MR1352863DOI10.1051/m2an/1995290505771
  49. 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

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