Combined finite element -- finite volume method (convergence analysis)

Mária Lukáčová-Medviďová

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 4, page 717-741
  • ISSN: 0010-2628

Abstract

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We present an efficient numerical method for solving viscous compressible fluid flows. The basic idea is to combine finite volume and finite element methods in an appropriate way. Thus nonlinear convective terms are discretized by the finite volume method over a finite volume mesh dual to a triangular grid. Diffusion terms are discretized by the conforming piecewise linear finite element method. In the paper we study theoretical properties of this scheme for the scalar nonlinear convection-diffusion equation. We prove the convergence of the numerical solution to the exact solution.

How to cite

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Lukáčová-Medviďová, Mária. "Combined finite element -- finite volume method (convergence analysis)." Commentationes Mathematicae Universitatis Carolinae 38.4 (1997): 717-741. <http://eudml.org/doc/248092>.

@article{Lukáčová1997,
abstract = {We present an efficient numerical method for solving viscous compressible fluid flows. The basic idea is to combine finite volume and finite element methods in an appropriate way. Thus nonlinear convective terms are discretized by the finite volume method over a finite volume mesh dual to a triangular grid. Diffusion terms are discretized by the conforming piecewise linear finite element method. In the paper we study theoretical properties of this scheme for the scalar nonlinear convection-diffusion equation. We prove the convergence of the numerical solution to the exact solution.},
author = {Lukáčová-Medviďová, Mária},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {compressible Navier-Stokes equations; nonlinear convection-diffusion equation; finite volume schemes; finite element method; numerical integration; apriori estimates; convergence of the scheme; nonlinear convection-diffusion equation; finite element method; finite volume scheme; a priori estimates; convergence of the numerical method},
language = {eng},
number = {4},
pages = {717-741},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Combined finite element -- finite volume method (convergence analysis)},
url = {http://eudml.org/doc/248092},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Lukáčová-Medviďová, Mária
TI - Combined finite element -- finite volume method (convergence analysis)
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 4
SP - 717
EP - 741
AB - We present an efficient numerical method for solving viscous compressible fluid flows. The basic idea is to combine finite volume and finite element methods in an appropriate way. Thus nonlinear convective terms are discretized by the finite volume method over a finite volume mesh dual to a triangular grid. Diffusion terms are discretized by the conforming piecewise linear finite element method. In the paper we study theoretical properties of this scheme for the scalar nonlinear convection-diffusion equation. We prove the convergence of the numerical solution to the exact solution.
LA - eng
KW - compressible Navier-Stokes equations; nonlinear convection-diffusion equation; finite volume schemes; finite element method; numerical integration; apriori estimates; convergence of the scheme; nonlinear convection-diffusion equation; finite element method; finite volume scheme; a priori estimates; convergence of the numerical method
UR - http://eudml.org/doc/248092
ER -

References

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  1. Adam D., Felgenhauer A., Roos H.-G., Stynes M., A nonconforming finite element method for a singularly perturbed boundary value problem, Computing 54 1 (1995), 1-25. (1995) Zbl0813.65105MR1314953
  2. Bardos C., LeRoux A.Y., Nedelec J.C., First order quasilinear equations with boundary conditions, Comm. in Part. Diff. Equa. 4 (9) (1979), 1017-1034. (1979) MR0542510
  3. Berger H., Feistauer M., Analysis of the finite element variational crimes in the numerical approximation of transonic flow, Math. Comput. 61 204 (1993), 493-521. (1993) Zbl0786.76051MR1192967
  4. Ciarlet P.G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979. Zbl0547.65072MR0520174
  5. Feistauer M., Mathematical Methods in Fluid Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 67, Longman Scientific & Technical, Harlow, 1993. Zbl0819.76001MR1266627
  6. Feistauer M., Felcman J., Dolejší V., Adaptive finite volume method for the numerical solution of the compressible Euler equations, in: S. Wagner, E.H. Hirschel, J. Périaux and R. Piva, Eds., {Computational Fluid Dynamics 94}, Vol. 2, Proc. of the Second European CFD Conference (John Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapore, 1994), pp. 894-901. 
  7. Feistauer M., Felcman J., Lukáčová-Medviďová M., Combined finite element-finite volume solution of compressible flow, Journal of Comput. and Appl. Math. 63 (1995), 179-199. (1995) MR1365559
  8. Feistauer M., Felcman J., Lukáčová-Medviďová M., On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems, Num. Methods for Part. Diff. Eqs. 13 (1997), 1-28. (1997) MR1436613
  9. Feistauer M., Knobloch P., Operator splitting method for compressible Euler and NavierStokes equations, Proc. of Internat. Workshop on Numerical Methods for the Navier-Stokes Equations, Heidelberg 1993, {Notes on Numerical Fluid Mechanics}, Vieweg, BraunschweigWiesbaden. 
  10. Felcman J., Finite volume solution of inviscid compressible fluid flow, ZAMM 71 (1991), 665-668. (1991) 
  11. Göhner U., Warnecke G., A shock indicator for adaptive transonic flow computations, Num. Math. 66 (1994), 423-448. (1994) MR1254397
  12. Göhner U., Warnecke G., A second order finite difference error indicator for adaptive transonic flow computations, Num. Math. 70 (1995), 129-161. (1995) MR1324735
  13. Ikeda T., Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Mathematics Studies 76, Lecture Notes in Numerical and Applied Analysis Vol. 4, North-Holland, Amsterdam-New York-Oxford, 1983. Zbl0508.65049MR0683102
  14. Lukáčová-Medviďová M., Numerical Solution of Compressible Flow, PhD Thesis, Fac. of Math. and Physics, Charles Univ., Prague, 1994. 
  15. Málek J., Nečas J., Rokyta M., Růžička M., Weak and Measure Valued Solutions to Evolutionary Partial Differential Equations, Applied Mathematics and Mathematical Computation 13, London, Chapman & Hall, 1996. 
  16. Ohmori K., Ushijima T., A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations, RAIRO Numer. Anal. 18 (1984), 309-322. (1984) Zbl0586.65080MR0751761
  17. Risch U., An upwind finite element method for singularly perturbed elliptic problems and local estimates in the L ∞ -norm, MAN 24 (1990), 235-264. (1990) MR1052149
  18. Schieweck F., Tobiska L., A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation, MAN 23 (1989), 627-647. (1989) Zbl0681.76032MR1025076
  19. Temam R., Navier-Stokes Equations, North-Holland, Amsterdam-New York-Oxford, 1977. Zbl1157.35333MR0609732
  20. Tobiska L., Full and weighted upwind finite element methods, in: Splines in Numerical Analysis (Mathematical Research Volume 52, J.W. Schmidt, H. Späth - eds.), Akademie-Verlag, Berlin, 1989. Zbl0685.65074MR1004263
  21. Vijayasundaram G., Transonic flow simulation using an upstream centered scheme of Godunov in finite elements, J. Comp. Phys. 63 (1986), 416-433. (1986) MR0835825

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