Journal of Computational Mathematics Vol.xx, No.x, 2021, 1–18. http://www.global-sci.org/jcm doi:10.4208/jcm.2011-m2019-0142 A NEW HYBRIDIZED MIXED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS * Abdelhamid Zaghdani University of Tunis, Boulevard du 9 avril 1939 Tunis, Department of Mathematics, Ensit, Taha Hussein Avenue, Montfleury, Tunis, Tunisia Northern Border University, Faculty of Arts and Science, Rafha, P.O 840, Saudi Arabia Email: hamido20042002@yahoo.fr Sayed Sayari Carthage University, Isteub, 2 Rue de l’Artisanat Charguia 2, 2035 Tunis, Tunisia Email: Sayari.sayed@gmail.com Miled EL Hajji 1) Department of Mathematics, Faculty of Sciences, University of Jeddah, Saudi Arabia ENIT-LAMSIN, BP. 37, 1002 Tunis-Belv´ ed` ere, Tunis El Manar University, Tunisia Email: miled.elhajji@enit.rnu.tn Abstract In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some opera- tors with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method. Mathematics subject classification: 65N30, 65N15, 35J20, 76S05, 35J46. Key words: Weak Galerkin, Weak gradient, Hybridized mixed finite element method, Second order elliptic problems. 1. Introduction Hybridized mixed finite element method for second order elliptic boundary value problems on polygonal domains provides approximations of the solution in terms of elementwise given scalar and vector valued functions and a multiplier on the set of interior edges or faces of the underlying triangulation of the domain. Recently, there is an attracted particular interest within a unified framework for hybridization of mixed and discontinuous Galerkin methods (see, e.g., [2, 3, 7, 8, 10–12, 14, 16, 17, 19, 20] and the references therein). In the literature, the hybridized mixed Galerkin method possesses local mass conservation and continuity of fluxes. That preserves some mathematical properties of physical systems in many applications such as modeling groundwater flow, reactive transport in porous media. That allowed to make a com- petitive numerical technique to provide a good approximation for both the velocity and pres- sure. Hybridized mixed Galerkin method was a clever implementation technique that contains more information than the solution obtained with numerical classical methods, it was described in [7]. Lately, a new perspective on hybridized mixed Galerkin method was developed and it * Received June 10, 2019 / Revised version received June 12, 2020 / Accepted November 17, 2020 / Published online March 10, 2021 / 1) Corresponding author