A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations

In this paper a hybridized weak Galerkin (HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced. The WG method uses weak functions and their weak derivatives which are defined as distributions. Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution. With this new feature, HWG method can be used to deal with jumps of the functions and their flux easily. Optimal order error estimate are established for the corresponding HWG finite element approximations for both {\color{black}primal variables} and the Lagrange multiplier. A Schur complement formulation of the HWG method is derived for implementation purpose. The validity of the theoretical results is demonstrated in numerical tests.Comment: 19 pages, 4 tables,it has been accepted for publication in SCIENCE CHINA Mathematics. arXiv admin note: substantial text overlap with arXiv:1402.1157, arXiv:1302.2707 by other author

Numerical simulation of combined mixing and separating flow in channel filled with porous media

Various flow bifurcations are investigated for two dimensional combined mixing and separating geometry. These consist of two reversed channel flows interacting through a gap in the common separating wall filled with porous media of Newtonian fluids and other with unidirectional fluid flows. The Steady solutions are obtained through an unsteady finite element approach that employs a Taylor-Galerkin/pressure-correction scheme. The influence of increasing inertia on flow rates are all studied. Close agreement is attained with numerical data in the porous channels for Newtonian fluids.Peer reviewedSubmitted Versio

A uniform and pressure-robust enriched Galerkin method for the Brinkman equations

This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. We derive, analyze, and compare two EG methods in this paper: standard and robust methods. The standard method requires a mesh size to be less than a viscous parameter to produce stable and accurate velocity solutions, which is impractical in the Darcy regime. Therefore, we propose the pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method yields error estimates independent of a pressure term and shows uniform performance from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We prove well-posedness and error estimates for both the standard and robust EG methods. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for the robust method