Convergence analysis for an exponentially fitted Finite Volume Method

Reiner Vanselow

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 6, page 1165-1188
  • ISSN: 0764-583X

Abstract

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The paper is devoted to the convergence analysis of a well-known cell-centered Finite Volume Method (FVM) for a convection-diffusion problem in ℝ 2 . This FVM is based on Voronoi boxes and exponential fitting. To prove the convergence of the FVM, we use a new nonconforming Petrov-Galerkin Finite Element Method (FEM) for which the system of linear equations coincides completely with that of the FVM. Thus, by proving convergence properties of the FEM we obtain similar ones for the FVM. For the error estimation of the FEM well-known statements have to be modified.

How to cite

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Vanselow, Reiner. "Convergence analysis for an exponentially fitted Finite Volume Method." ESAIM: Mathematical Modelling and Numerical Analysis 34.6 (2010): 1165-1188. <http://eudml.org/doc/197584>.

@article{Vanselow2010,
abstract = { The paper is devoted to the convergence analysis of a well-known cell-centered Finite Volume Method (FVM) for a convection-diffusion problem in $\mathbb\{R\}^2$. This FVM is based on Voronoi boxes and exponential fitting. To prove the convergence of the FVM, we use a new nonconforming Petrov-Galerkin Finite Element Method (FEM) for which the system of linear equations coincides completely with that of the FVM. Thus, by proving convergence properties of the FEM we obtain similar ones for the FVM. For the error estimation of the FEM well-known statements have to be modified. },
author = {Vanselow, Reiner},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Convection-diffusion problem; cell-centered finite volume method; Voronoi boxes; exponential fitting; convergence analysis; nonconforming finite element method.; convergence; finite volume method; convection-diffusion problem; Voronoi box; Petrov-Galerkin finite element method},
language = {eng},
month = {3},
number = {6},
pages = {1165-1188},
publisher = {EDP Sciences},
title = {Convergence analysis for an exponentially fitted Finite Volume Method},
url = {http://eudml.org/doc/197584},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Vanselow, Reiner
TI - Convergence analysis for an exponentially fitted Finite Volume Method
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 6
SP - 1165
EP - 1188
AB - The paper is devoted to the convergence analysis of a well-known cell-centered Finite Volume Method (FVM) for a convection-diffusion problem in $\mathbb{R}^2$. This FVM is based on Voronoi boxes and exponential fitting. To prove the convergence of the FVM, we use a new nonconforming Petrov-Galerkin Finite Element Method (FEM) for which the system of linear equations coincides completely with that of the FVM. Thus, by proving convergence properties of the FEM we obtain similar ones for the FVM. For the error estimation of the FEM well-known statements have to be modified.
LA - eng
KW - Convection-diffusion problem; cell-centered finite volume method; Voronoi boxes; exponential fitting; convergence analysis; nonconforming finite element method.; convergence; finite volume method; convection-diffusion problem; Voronoi box; Petrov-Galerkin finite element method
UR - http://eudml.org/doc/197584
ER -

References

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  1. L. Angermann, Error Estimate for the Finite-Element Solution of an Elliptic Singularly Perturbed Problem. IMA J. Numer. Anal.15 (1995) 161-196.  
  2. R.E. Bank, J.F. Bürgler, W. Fichtner and R.K. Smith, Some Upwinding Techniques for Finite Element Approximations of Convection-Diffusion Equations. Numer. Math.58 (1990) 185-202.  
  3. R.E. Bank, W.M. Jr. Coughran and L.C. Cowsar, The Finite Volume Scharfetter-Gummel Method for Steady Convection Diffusion Equations. Comput. Visual Sci.1 (1998) 123-136.  
  4. J. Baranger, J.-F. Maitre and F. Oudin, Connection between Finite Volume and Mixed Finite Element Methods. RAIRO Modél. Math. Anal. Numér.30 (1996) 445-465.  
  5. D. Braess, Finite Elemente. Springer, Berlin (1992).  
  6. P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in Handbook of Numerical Analysis, Vol. II, Part 1, P.G. Ciarlet and J.L. Lions Eds., Elsevier, Amsterdam (1991) 17-351.  
  7. R. Eymard, T. Gallouet and R. Herbin, Convergence of Finite Volume Schemes for Semilinear Convection Diffusion Equations. Numer. Math.1 (1999) 1-26.  
  8. E. Gatti, S. Micheletti and R. Sacco, A New Galerkin Framework for the Drift-Diffusion Equation in Semiconductors. East-West J. Numer. Math.6 (1998) 101-135.  
  9. B. Heinrich, Finite Difference Methods on Irregular Networks. A Generalized Approach to Second Order Problems. Akademie, Berlin (1987).  
  10. R. Herbin, An Error Estimate for a Finite Volume Scheme for a Diffusion-Convection Problem on a Triangular Mesh. Numer. Methods Partial Differential Equations11 (1995) 165-173.  
  11. R.D. Lazarov and I.D. Mishev, Finite Volume Methods for Reaction-Diffusion Problems, in Finite Volumes for Complex Applications, F. Benkhaldoun and R. Vilsmeier Eds., Hermes, Paris (1996) 231-240.  
  12. J.J.H. Miller and S. Wang, A New Non-Conforming Petrov-Galerkin Finite Element Method with Triangular Elements for an Advection-Diffusion Problem. IMA J. Numer. Anal.14 (1994) 257-276.  
  13. I.D. Mishev, Finite Volume and Finite Volume Element Methods for Nonsymmetric Problems. Ph.D. thesis, Texas A&M University (1996).  
  14. K.W. Morton, Numerical Solution of Convection-Diffusion Problems. Chapman and Hall, London (1996).  
  15. K.W. Morton, M. Stynes and E. Süli, Analysis of a Cell-Vertex Finite Volume Method for Convection-Diffusion Problems. Math. Comp.66 (1997) 1369-1406.  
  16. H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Springer, London (1996).  
  17. R. Sacco and M. Stynes, Finite Element Methods for Convection-Diffusion Problems Using Exponential Splines on Triangles. Comput. Math. Appl.35 (1998) 35-45.  
  18. R. Sacco, E. Gatti and L. Gotusso, A Nonconforming Exponentially Fitted Finite Element Method for Two-Dimensional Drift-Diffusion Models in Semiconductors. Numer. Methods Partial Differential Equations15 (1999) 133-150.  
  19. H.-P. Scheffler and R. Vanselow, Convergence Analysis of a Cell-Centered FVM, in Finite Volumes for Complex Applications II, R. Vilsmeier, F. Benkhaldoun and D. Hänel Eds., Hermes, Paris (1999) 181-188.  
  20. L.L. Schumaker, Spline Functions: Basic Theory. Wiley, New York (1981).  
  21. S. Selberherr, Analysis and Simulation of Semiconductor Devices. Springer, Wien (1984).  
  22. G. Strang, Variational Crimes in the Finite Element Method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz Ed., Academic Press (1972) 689-710.  
  23. R. Vanselow and H.-P. Scheffler, Convergence Analysis of a Finite Volume Method via a New Nonconforming Finite Element Method. Numer. Methods Partial Differential Equations14 (1998) 213-231.  
  24. R. Vanselow, Convergence Analysis for an Exponentially Fitted FVM. Preprint MATH-NM-09-99, TU Dresden (1999).  
  25. J. Xu and L. Zikatanov, A Monotone Finite Element Scheme for Convection-Diffusion Equations. Math. Comp.68 (1999) 1429-1446.  

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