# 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.  Zbl0829.65127
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.  Zbl0880.65084
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.  Zbl0567.65078
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.  Zbl1008.65080
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.  Zbl0857.65116
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.  Zbl0871.76040
7. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag (1991).  Zbl0788.73002
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.  Zbl0599.65072
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.  Zbl0631.65107
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.  Zbl0689.65065
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.  Zbl0957.65099
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.  Zbl0941.76050
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.  Zbl0987.65111
14. Z. Chen, Expanded mixed finite element methods for linear second-order elliptic problems I. RAIRO Modél. Math. Anal. Numér.32 (1998) 479–499.  Zbl0910.65079
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.  Zbl0986.65110
16. Z. Chen and J. Douglas, Prismatic mixed finite elements for second order elliptic problems. Calcolo26 (1989) 135–148.  Zbl0711.65089
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.  Zbl0924.65099
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.  Zbl0914.65107
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 Zbl1007.65091
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.  Zbl1015.65068
21. P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).  Zbl0383.65058
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.  Zbl0927.65118
23. B. Courbet and J.P. Croisille, Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér.32 (1998) 631–649.  Zbl0920.65065
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.  Zbl1035.65123
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.  Zbl0698.65060
28. R.S. Falk and J.E. Osborn, Error estimates for mixed methods. RAIRO Anal. Numér.14 (1980) 249–277.  Zbl0467.65062
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.  Zbl1007.65104
30. J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math.95 (2003) 527–550.  Zbl1044.65084
31. S. Micheletti and R. Sacco, Dual-primal mixed finite elements for elliptic problems. Numer. Methods Partial Differential Equations17 (2001) 137–151.  Zbl0979.65103
32. J.C. Nedelec, Mixed finite elements in ${ℝ}^{3}$. Numer. Math.35 (1980) 315–341.  Zbl0419.65069
33. J.C. Nedelec, A new family of mixed finite elements in ${ℝ}^{3}$. Numer. Math.50 (1986) 57–81.  Zbl0625.65107
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.  Zbl1001.76060
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.  Zbl0362.65089
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.  Zbl1010.65045
37. J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland (1991) 523–639.  Zbl0875.65090
38. R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation. Numer. Methods Partial Differential Equations13 (1997) 215–236.  Zbl0890.65132
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.  Zbl0644.65062

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