New mixed finite volume methods for second order eliptic problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 1, pp. 123-147.

In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problems which are based on $H\left(\mathrm{div}\right)$-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.

DOI : https://doi.org/10.1051/m2an:2006001
Classification : 65F10,  65N15,  65N30
Mots clés : mixed method, finite volume method, discontinuous finite element method, conservative method
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Kim, Kwang Y. New mixed finite volume methods for second order eliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 1, pp. 123-147. doi : 10.1051/m2an:2006001. http://www.numdam.org/articles/10.1051/m2an:2006001/

[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. | Zbl 0829.65127

[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. | Zbl 0880.65084

[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. | Numdam | Zbl 0567.65078

[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. | Zbl 1008.65080

[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. | Numdam | Zbl 0857.65116

[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. | Zbl 0871.76040

[7] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[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. | Zbl 0599.65072

[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. | Zbl 0631.65107

[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. | Numdam | Zbl 0689.65065

[11] F. Brezzi, G. Manzini, L.D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations 16 (2000) 365-378. | Zbl 0957.65099

[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. | Zbl 0941.76050

[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. | Zbl 0987.65111

[14] Z. Chen, Expanded mixed finite element methods for linear second-order elliptic problems I. RAIRO Modél. Math. Anal. Numér. 32 (1998) 479-499. | Numdam | Zbl 0910.65079

[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. | Zbl 0986.65110

[16] Z. Chen and J. Douglas, Prismatic mixed finite elements for second order elliptic problems. Calcolo 26 (1989) 135-148. | Zbl 0711.65089

[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. | Zbl 0924.65099

[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. | Zbl 0914.65107

[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 | Zbl 1007.65091

[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. | Zbl 1015.65068

[21] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058

[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. | Zbl 0927.65118

[23] B. Courbet and J.P. Croisille, Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér. 32 (1998) 631-649. | Numdam | Zbl 0920.65065

[24] J.P. Croisille, Finite volume box schemes and mixed methods ESAIM: M2AN 34 (2000) 1087-1106. | Numdam | Zbl 0966.65082

[25] J.P. Croisille and I. Greff, Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations 18 (2002) 355-373. | Zbl 1004.65113

[26] C. Dawson, The ${𝒫}^{K+1}-{𝒮}^{K}$ local discontinuous Galerkin method for elliptic equations. SIAM J. Numer. Anal. 40 (2002) 2151-2170. | Zbl 1035.65123

[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. | Numdam | Zbl 0698.65060

[28] R.S. Falk and J.E. Osborn, Error estimates for mixed methods. RAIRO Anal. Numér. 14 (1980) 249-277. | Numdam | Zbl 0467.65062

[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. | Zbl 1007.65104

[30] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527-550. | Zbl 1044.65084

[31] S. Micheletti and R. Sacco, Dual-primal mixed finite elements for elliptic problems. Numer. Methods Partial Differential Equations 17 (2001) 137-151. | Zbl 0979.65103

[32] J.C. Nedelec, Mixed finite elements in ${ℝ}^{3}$. Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[33] J.C. Nedelec, A new family of mixed finite elements in ${ℝ}^{3}$. Numer. Math. 50 (1986) 57-81. | Zbl 0625.65107

[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. | Zbl 1001.76060

[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. | Zbl 0362.65089

[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. | Zbl 1010.65045

[37] J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland (1991) 523-639. | Zbl 0875.65090

[38] R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation. Numer. Methods Partial Differential Equations 13 (1997) 215-236. | Zbl 0890.65132

[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. | Zbl 0644.65062

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