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.

Keywords: mixed method, finite volume method, discontinuous finite element method, conservative method

@article{M2AN_2006__40_1_123_0, author = {Kim, Kwang Y.}, title = {New mixed finite volume methods for second order eliptic problems}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {123--147}, publisher = {EDP-Sciences}, volume = {40}, number = {1}, year = {2006}, doi = {10.1051/m2an:2006001}, mrnumber = {2223507}, zbl = {1097.65116}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2006001/} }

TY - JOUR AU - Kim, Kwang Y. TI - New mixed finite volume methods for second order eliptic problems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 123 EP - 147 VL - 40 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006001/ DO - 10.1051/m2an:2006001 LA - en ID - M2AN_2006__40_1_123_0 ER -

%0 Journal Article %A Kim, Kwang Y. %T New mixed finite volume methods for second order eliptic problems %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 123-147 %V 40 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2006001/ %R 10.1051/m2an:2006001 %G en %F M2AN_2006__40_1_123_0

Kim, Kwang Y. New mixed finite volume methods for second order eliptic problems. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 1, pp. 123-147. doi : 10.1051/m2an:2006001. http://archive.numdam.org/articles/10.1051/m2an:2006001/

[1] On the implementation of mixed methods as nonconforming methods for second order elliptic problems. Math. Comp. 64 (1995) 943-972. | Zbl

and ,[2] Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828-852. | Zbl

, and ,[3] Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7-32. | Numdam | Zbl

and ,[4] Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779. | Zbl

, , and ,[5] Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 445-465. | Numdam | Zbl

, and ,[6] 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

and ,[7] Mixed and hybrid finite element methods. Springer-Verlag (1991). | MR | Zbl

and ,[8] Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | Zbl

, and ,[9] Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237-250. | Zbl

, , and ,[10] Efficient rectangular mixed finite elements in two and three variables. RAIRO Modél. Math. Anal. Numér. 21 (1987) 581-604. | Numdam | Zbl

, , and ,[11] Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations 16 (2000) 365-378. | Zbl

, , , and ,[12] Control-volume mixed finite element Methods. Comput. Geosci. 1 (1997) 289-315. | Zbl

, , and ,[13] An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676-1706. | Zbl

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

,[15] On the relationship of various discontinuous finite element methods for second-order elliptic equations. East-West J. Numer. Math. 9 (2001) 99-122. | Zbl

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

and ,[17] A general mixed covolume framework for constructing conservative schemes for elliptic problems. Math. Comp. 68 (1999) 991-1011. | Zbl

and ,[18] Mixed covolume methods for elliptic problems on triangular grids. SIAM J. Numer. Anal. 35 (1998) 1850-1861. | Zbl

, and ,[19] 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

, and ,[20] Mixed finite volume methods on non-staggered quadrilateral grids for elliptic problems. Math. Comp. 72 (2003) 525-539. | Zbl

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

,[22] The local discontinuous Galerkin method for time-dependent convection-diffusion system. SIAM J. Numer. Anal. 35 (1998) 2440-2463. | Zbl

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

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

,[25] Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations 18 (2002) 355-373. | Zbl

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

,[27] 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

,[28] Error estimates for mixed methods. RAIRO Anal. Numér. 14 (1980) 249-277. | Numdam | Zbl

and ,[29] Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1343-1365. | Zbl

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

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

and ,[32] Mixed finite elements in ${\mathbb{R}}^{3}$. Numer. Math. 35 (1980) 315-341. | Zbl

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

,[34] An $hp$-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17 (2002) 561-571. | Zbl

and ,[35] 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

and ,[36] 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

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

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

and ,[39] On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. | Zbl

and ,*Cited by Sources: *