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 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 can be taken as the square of the coarse mesh size , or a higher order power with a proper choice of parameter . 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 are also given. In addition, our approach is shown to provide an asymptotically exact a posteriori error estimator for the primary variable in norm.
Accepté le :
DOI : 10.1051/m2an/2016062
Mots clés : Finite element method, elliptic problem, flux variable, mixed finite element
@article{M2AN_2017__51_4_1303_0, author = {Ku, JaEun and Lee, Young Ju and Sheen, Dongwoo}, title = {A hybrid two-step finite element method for flux approximation: a priori estimates}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1303--1316}, publisher = {EDP-Sciences}, volume = {51}, number = {4}, year = {2017}, doi = {10.1051/m2an/2016062}, mrnumber = {3702414}, zbl = {1379.65089}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016062/} }
TY - JOUR AU - Ku, JaEun AU - Lee, Young Ju AU - Sheen, Dongwoo TI - A hybrid two-step finite element method for flux approximation: a priori estimates JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1303 EP - 1316 VL - 51 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016062/ DO - 10.1051/m2an/2016062 LA - en ID - M2AN_2017__51_4_1303_0 ER -
%0 Journal Article %A Ku, JaEun %A Lee, Young Ju %A Sheen, Dongwoo %T A hybrid two-step finite element method for flux approximation: a priori estimates %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1303-1316 %V 51 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016062/ %R 10.1051/m2an/2016062 %G en %F M2AN_2017__51_4_1303_0
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/
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