A hybrid two-step finite element method for flux approximation: a priori estimates
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1303-1316.

We present a new two–step method based on the hybridization of mesh sizes in the traditional mixed finite element method. On a coarse mesh, the primary variable is approximated by a standard Galerkin method, whose computational cost is very low. Then, on a fine mesh, an H(div) projection of the dual variable is sought as an accurate approximation for the flux variable. Our method does not rely on the framework of traditional mixed formulations, the choice of pair of finite element spaces is, therefore, free from the requirement of inf-sup stability condition. More precisely, our method is formulated in a fully decoupled manner, still achieving an optimal error convergence order. This leads to a computational strategy much easier and wider to implement than the mixed finite element method. Additionally, the independently posed solution strategy allows to use different meshes as well as different discretization schemes in the calculation of the primary and flux variables. We show that the finer mesh size h can be taken as the square of the coarse mesh size H, or a higher order power with a proper choice of parameter δ. This means that the computational cost for the coarse-grid solution is negligible compared to that for the fine-grid solution. In fact, numerical experiments show an advantage of using our strategy compared to the mixed finite element method. Some guidelines to choose an optimal parameter δ are also given. In addition, our approach is shown to provide an asymptotically exact a posteriori error estimator for the primary variable p in H 1 norm.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016062
Classification : 65N30, 65N15
Mots clés : Finite element method, elliptic problem, flux variable, mixed finite element
Ku, JaEun 1 ; Lee, Young Ju 2 ; Sheen, Dongwoo 3

1 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA.
2 Department of Mathematics, Texas State University, San Marcos, TX 78666, USA.
3 Department of Mathematics, Seoul National University, Seoul 08826, Korea.
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     title = {A hybrid two-step finite element method for flux approximation: a priori estimates},
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Ku, JaEun; Lee, Young Ju; Sheen, Dongwoo. A hybrid two-step finite element method for flux approximation: a priori estimates. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1303-1316. doi : 10.1051/m2an/2016062. http://www.numdam.org/articles/10.1051/m2an/2016062/

T. Arbogast, M.F. 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. | DOI | MR | Zbl

D.N. Arnold, R.S. Falk and R. Winther, Multigrid in H(div) and H(curl). Numer. Math. 85 (2000) 197–218. | DOI | MR | Zbl

D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Springer (2013). | MR | Zbl

A. Brandt, Algebraic multigrid theory: The symmetric case. Appl. Math. Comput. 19 (1986) 23–56. | MR | Zbl

J.H. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68 (1994) 311–324. | DOI | MR | Zbl

J.H. Brandts, Superconvergence for triangular order k=1 Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods. Appl. Numer. Math. 34 (2000) 39–58. | DOI | MR | Zbl

S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Number 15 in Texts in Applied Mathematics. 2nd edition. Springer Verlag, New York (2002). | MR | Zbl

F. Brezzi, J. Douglas Jr., R. Duran and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. | DOI | MR | Zbl

F. Brezzi, J. Douglas, Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. | DOI | MR | Zbl

M. Cai, M. Mu and J. Xu, Numerical solution to a mixed Navier-Sokes/Darcy model by the two-grid approach. SIAM J. Numer. Anal. 47 (2009) 3325–3338. | DOI | MR | Zbl

Z. Cai, V. Carey, J. Ku and E.-J. Park, Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global L 2 norm. Comput. Math. Appl. 70 (2015) 648–659. | DOI | MR | Zbl

Z. Cai and S. Zhang, Flux recovery and a posteriori error estimators: Conforming elements for scalar elliptic equations. SIAM J. Numer. Anal. 48 (2010) 578–602. | DOI | MR | Zbl

J. Douglas, Jr. and J.E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media. Math. Comput. 41 (1983) 441–459. | DOI | MR | Zbl

J. Douglas, Jr and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44 (1985) 39–52. | DOI | MR | Zbl

R.E. Ewing, M. Liu and J. Wang, A new superconvergence for mixed finite element approximations. SIAM J. Numer. Anal. 40 (2002) 2133–2150. | DOI | MR | Zbl

V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes equations. Theory and algorithm. Springer Verlag, Berlin (1986). | MR | Zbl

R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45 (2007) 2483–2509. | DOI | MR | Zbl

J. Jin, S. Shu and J. Xu, A two-grid discretization method for decoupling systems of partial differential equations. Math. Comput. 75 (2006) 1617–1626. | DOI | MR | Zbl

J. Ku, Supercloseness of the mixed finite element method for the primary function on unstructured meshes and its applications. BIT 54 (2014) 1087–1097. | DOI | MR | Zbl

Y.-J. Lee and J. Xu, New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1180–1206. | DOI | MR | Zbl

J. Nédélec, Mixed finite elements in R 3 . Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

P. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems. In Proc. of Conference on FEM held in Rome in 1975. Vol. 606 of Lect. Notes Math. Springer Verlag (1977). | MR | Zbl

J. Wang, Superconvergence and extrapolation for mixed element methods on rectangular domains. Math. Comput. 56 (1991) 477–503. | DOI | MR | Zbl

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