Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T09:28:16.430Z Has data issue: false hasContentIssue false
  • English
  • Français

New mixed finite volume methods for second ordereliptic problems

Published online by Cambridge University Press:  23 February 2006

Kwang Y. Kim
Kwang Y. Kim*
Affiliation:
Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701 South Korea. toheart@acoustic.kaist.ac.kr
Get access

Abstract

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

Keywords

Mixed methodfinite volume methoddiscontinuous finite element methodconservative method.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 1 , January 2006 , pp. 123 - 147
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arbogast, T. and Chen, Z., On the implementation of mixed methods as nonconforming methods for second order elliptic problems. Math. Comp. 64 (1995) 943–972.
Arbogast, T., Wheeler, M. and Yotov, I., Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828–852. CrossRef
Arnold, D.N. and Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7–32. CrossRef
Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. CrossRef
Baranger, J., Maître, J.F. and Oudin, F., Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 445–465. CrossRef
Bassi, F. and Rebay, S., 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. CrossRef
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag (1991).
Brezzi, F., Douglas, J. and Marini, L.D., Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. CrossRef
Brezzi, F., Douglas, J., Durán, R. and Fortin, M., Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. CrossRef
Brezzi, F., Douglas, J., Fortin, M. and Marini, L.D., Efficient rectangular mixed finite elements in two and three variables. RAIRO Modél. Math. Anal. Numér. 21 (1987) 581–604. CrossRef
Brezzi, F., Manzini, G., Marini, L.D., Pietra, P. and Russo, A., Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations 16 (2000) 365–378. 3.0.CO;2-Y>CrossRef
Cai, Z., Jones, J.E., McCormick, S.F. and Russell, T.F., Control-volume mixed finite element Methods. Comput. Geosci. 1 (1997) 289–315. CrossRef
Castillo, P., Cockburn, B., Perugia, I. and Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676–1706. CrossRef
Chen, Z., Expanded mixed finite element methods for linear second-order elliptic problems I. RAIRO Modél. Math. Anal. Numér. 32 (1998) 479–499. CrossRef
Chen, Z., On the relationship of various discontinuous finite element methods for second-order elliptic equations. East-West J. Numer. Math. 9 (2001) 99–122.
Chen, Z. and Douglas, J., Prismatic mixed finite elements for second order elliptic problems. Calcolo 26 (1989) 135–148. CrossRef
Chou, S.H. and Vassilevski, P.S., A general mixed covolume framework for constructing conservative schemes for elliptic problems. Math. Comp. 68 (1999) 991–1011. CrossRef
Chou, S.H., Kwak, D.Y. and Vassilevski, P., Mixed covolume methods for elliptic problems on triangular grids. SIAM J. Numer. Anal. 35 (1998) 1850–1861. CrossRef
Chou, S.H., Kwak, D.Y. and Kim, K.Y., 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 CrossRef
Chou, S.H., Kwak, D.Y. and Kim, K.Y., Mixed finite volume methods on non-staggered quadrilateral grids for elliptic problems. Math. Comp. 72 (2003) 525–539. CrossRef
P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).
Cockburn, B. and Shu, C.W., The local discontinuous Galerkin method for time-dependent convection-diffusion system. SIAM J. Numer. Anal. 35 (1998) 2440–2463. CrossRef
Courbet, B. and Croisille, J.P., Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér. 32 (1998) 631–649. CrossRef
J.P. Croisille, Finite volume box schemes and mixed methods ESAIM: M2AN 34 (2000) 1087–1106.
Croisille, J.P. and Greff, I., Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations 18 (2002) 355–373. CrossRef
Dawson, C., The $\mathcal{P}^{K+1}-\mathcal{S}^K$ local discontinuous Galerkin method for elliptic equations. SIAM J. Numer. Anal. 40 (2002) 2151–2170. CrossRef
Durán, R.G., 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. CrossRef
Falk, R.S. and Osborn, J.E., Error estimates for mixed methods. RAIRO Anal. Numér. 14 (1980) 249–277.
Feng, X. and Karakashian, O.A., Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1343–1365. CrossRef
Gopalakrishnan, J. and Kanschat, G., A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527–550. CrossRef
Micheletti, S. and Sacco, R., Dual-primal mixed finite elements for elliptic problems. Numer. Methods Partial Differential Equations 17 (2001) 137–151. 3.0.CO;2-0>CrossRef
Nedelec, J.C., Mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 35 (1980) 315–341. CrossRef
Nedelec, J.C., A new family of mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 50 (1986) 57–81. CrossRef
Perugia, I. and Schötzau, D., An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17 (2002) 561–571. CrossRef
Raviart, P.A. and Thomas, J.M., 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. CrossRef
Riviere, B., Wheeler, M.F. and Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 902–931. CrossRef
J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland (1991) 523–639.
Sacco, R. and Saleri, F., Mixed finite volume methods for semiconductor device simulation. Numer. Methods Partial Differential Equations 13 (1997) 215–236. 3.0.CO;2-Q>CrossRef
Weiser, A. and Wheeler, M.F., On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351–375. CrossRef