# New mixed finite volume methods for second order eliptic problems

• Volume: 40, Issue: 1, page 123-147
• ISSN: 0764-583X

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## Abstract

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In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problems which are based on H(div)-conforming approximations for the vector variable and discontinuous approximations for the scalar variable. The discretization is fulfilled by combining the ideas of the traditional finite volume box method and the local discontinuous Galerkin method. We propose two different types of methods, called Methods I and II, and show that they have distinct advantages over the mixed methods used previously. In particular, a clever elimination of the vector variable leads to a primal formulation for the scalar variable which closely resembles discontinuous finite element methods. We establish error estimates for these methods that are optimal for the scalar variable in both methods and for the vector variable in Method II.

## How to cite

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Kim, Kwang Y.. "New mixed finite volume methods for second order eliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis 40.1 (2006): 123-147. <http://eudml.org/doc/249758>.

@article{Kim2006,
abstract = { In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problems which are based on H(div)-conforming approximations for the vector variable and discontinuous approximations for the scalar variable. The discretization is fulfilled by combining the ideas of the traditional finite volume box method and the local discontinuous Galerkin method. We propose two different types of methods, called Methods I and II, and show that they have distinct advantages over the mixed methods used previously. In particular, a clever elimination of the vector variable leads to a primal formulation for the scalar variable which closely resembles discontinuous finite element methods. We establish error estimates for these methods that are optimal for the scalar variable in both methods and for the vector variable in Method II. },
author = {Kim, Kwang Y.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mixed method; finite volume method; discontinuous finite element method; conservative method.; conservative method; mixed finite volume methods; local discontinuous Galerkin method; mixed finite element methods; error estimates},
language = {eng},
month = {2},
number = {1},
pages = {123-147},
publisher = {EDP Sciences},
title = {New mixed finite volume methods for second order eliptic problems},
url = {http://eudml.org/doc/249758},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Kim, Kwang Y.
TI - New mixed finite volume methods for second order eliptic problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/2//
PB - EDP Sciences
VL - 40
IS - 1
SP - 123
EP - 147
AB - In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problems which are based on H(div)-conforming approximations for the vector variable and discontinuous approximations for the scalar variable. The discretization is fulfilled by combining the ideas of the traditional finite volume box method and the local discontinuous Galerkin method. We propose two different types of methods, called Methods I and II, and show that they have distinct advantages over the mixed methods used previously. In particular, a clever elimination of the vector variable leads to a primal formulation for the scalar variable which closely resembles discontinuous finite element methods. We establish error estimates for these methods that are optimal for the scalar variable in both methods and for the vector variable in Method II.
LA - eng
KW - Mixed method; finite volume method; discontinuous finite element method; conservative method.; conservative method; mixed finite volume methods; local discontinuous Galerkin method; mixed finite element methods; error estimates
UR - http://eudml.org/doc/249758
ER -

## References

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1. T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems. Math. Comp.64 (1995) 943–972.
2. T. Arbogast, M. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal.34 (1997) 828–852.
3. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér.19 (1985) 7–32.
4. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779.
5. J. Baranger, J.F. Maître and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér.30 (1996) 445–465.
6. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys.131 (1997) 267–279.
7. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag (1991).
8. F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.
9. F. Brezzi, J. Douglas, R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math.51 (1987) 237–250.
10. F. Brezzi, J. Douglas, M. Fortin and L.D. Marini, Efficient rectangular mixed finite elements in two and three variables. RAIRO Modél. Math. Anal. Numér.21 (1987) 581–604.
11. F. Brezzi, G. Manzini, L.D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations16 (2000) 365–378.
12. Z. Cai, J.E. Jones, S.F. McCormick and T.F. Russell, Control-volume mixed finite element Methods. Comput. Geosci.1 (1997) 289–315.
13. P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal.38 (2000) 1676–1706.
14. Z. Chen, Expanded mixed finite element methods for linear second-order elliptic problems I. RAIRO Modél. Math. Anal. Numér.32 (1998) 479–499.
15. Z. Chen, On the relationship of various discontinuous finite element methods for second-order elliptic equations. East-West J. Numer. Math.9 (2001) 99–122.
16. Z. Chen and J. Douglas, Prismatic mixed finite elements for second order elliptic problems. Calcolo26 (1989) 135–148.
17. S.H. Chou and P.S. Vassilevski, A general mixed covolume framework for constructing conservative schemes for elliptic problems. Math. Comp.68 (1999) 991–1011.
18. S.H. Chou, D.Y. Kwak and P. Vassilevski, Mixed covolume methods for elliptic problems on triangular grids. SIAM J. Numer. Anal.35 (1998) 1850–1861.
19. S.H. Chou, D.Y. Kwak and K.Y. Kim, A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: the overlapping covolume case. SIAM J. Numer. Anal.39 (2001) 1170–1196
20. S.H. Chou, D.Y. Kwak and K.Y. Kim, Mixed finite volume methods on non-staggered quadrilateral grids for elliptic problems. Math. Comp.72 (2003) 525–539.
21. P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).
22. B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion system. SIAM J. Numer. Anal.35 (1998) 2440–2463.
23. B. Courbet and J.P. Croisille, Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér.32 (1998) 631–649.
24. J.P. Croisille, Finite volume box schemes and mixed methods ESAIM: M2AN34 (2000) 1087–1106.
25. J.P. Croisille and I. Greff, Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations18 (2002) 355–373.
26. C. Dawson, The ${𝒫}^{K+1}-{𝒮}^{K}$ local discontinuous Galerkin method for elliptic equations. SIAM J. Numer. Anal.40 (2002) 2151–2170.
27. R.G. Durán, Error analysis in ${L}^{p},1\le p\le \infty$, for mixed finite element methods for linear and quasi-linear elliptic problems. RAIRO Modél. Math. Anal. Numér.22 (1988) 371–387.
28. R.S. Falk and J.E. Osborn, Error estimates for mixed methods. RAIRO Anal. Numér.14 (1980) 249–277.
29. X. Feng and O.A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal.39 (2001) 1343–1365.
30. J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math.95 (2003) 527–550.
31. S. Micheletti and R. Sacco, Dual-primal mixed finite elements for elliptic problems. Numer. Methods Partial Differential Equations17 (2001) 137–151.
32. J.C. Nedelec, Mixed finite elements in ${ℝ}^{3}$. Numer. Math.35 (1980) 315–341.
33. J.C. Nedelec, A new family of mixed finite elements in ${ℝ}^{3}$. Numer. Math.50 (1986) 57–81.
34. I. Perugia and D. Schötzau, An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput.17 (2002) 561–571.
35. P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Proc.Conference on Mathematical Aspects of Finite Element Methods, Springer-Verlag. Lect. Notes Math.606 (1977) 292–315.
36. B. Riviere, M.F. Wheeler and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal.39 (2001) 902–931.
37. J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland (1991) 523–639.
38. R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation. Numer. Methods Partial Differential Equations13 (1997) 215–236.
39. A. Weiser and M.F. Wheeler, On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal.25 (1988) 351–375.

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