A hybrid two-step finite element method for flux approximation: a priori estimates

¹ Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA.
jku@math.okstate.edu
² Department of Mathematics, Texas State University, San Marcos, TX 78666, USA.
yjlee@txstate.edu
³ Department of Mathematics, Seoul National University, Seoul 08826, Korea.
dongwoosheen@gmail.com

Received: 7 December 2014
Revised: 15 May 2016
Accepted: 8 September 2016

Abstract

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 H¹ norm.