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In this article, we analyze a quadratic equal-order stabilized finite element approximation for the incompressible Stokes equations based on two local Gauss integrations. Our method only offsets the discrete pressure gradient space by the residual of the simple and symmetry term at element level to circumvent the inf-sup condition. And this method does not require specification of a stabilization parameter, and always leads to a symmetric linear system. Furthermore, this method is unconditionally stable, and can be implemented at the element level with minimal additional cost. Finally, we give some numerical simulations to show good stability and accuracy properties of the method.

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In this paper, we propose and analyze two stabilized mixed finite element methods for the dual-porosity-Stokes model, which couples the free flow region and microfracture-matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual-porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. KEYWORDS decoupled numerical methods, dual-porosity-Stokes model, horizontal wellbore, mixed finite elements, stabilization Int J Numer Methods Eng. 2019;120:803-833. wileyonlinelibrary.com/journal/nme 804 AL MAHBUB ET AL.model, It is not surprising that a great deal of effort has been devoted to develop appropriate numerical methods to solve the (Navier-)Stokes-Darcy fluid flow system, including coupled finite element methods, 17-21 domain decomposition methods, 22-31 Lagrange multiplier methods, 3,32,33 mortar finite element methods, 34,35 least-square methods, 36-38 partitioned time-stepping methods, 39,40 two-grid and multigrid methods, 41-44 discontinuous Galerkin finite element methods, 45-50 boundary integral methods, 51,52 and many others. [53][54][55][56][57][58] Although the traditional Stokes-Darcy model has been well studied, it has limitation to describe the heterogeneity of the porous medium which contains multiple porosities. It is worth to notice that the realistic naturally fractured reservoir consists of two coexisting and interacting medium, namely, tight matrix and microfractures. Furthermore, it is more important to characterize the intrinsic properties and accurately modeled the flow interactions between each medium. 59,60 The first multiporosity model was proposed by Barenblatt et al 61 for the naturally fractured reservoir in 1960 where the microfracture and matrix systems are formulated by individual but overlapping continua. Warren and Root 62 developed a homogeneous orthotropic dual-porosity model in 1963 based on the model proposed by Barenblatt. There are many applications for the dual-porosity model such as the geothermal system, hydrogeology, pe...

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In this article, supposing that the velocity, pressure, and temperature are approximated by the elements P2−P1−P2, and applying the orthogonal projection technique, we introduce two Gauss integrations as a stabilizing term in the common variational multiscale (VMS) method and derive a new VMS (Two Gauss VMS) method for steady‐state natural convection problem. Comparing with the common VMS method, the Two Gauss VMS method does not need to introduce any extra variable and reduces the degrees of freedom of the discrete system a lot, but gets the same stabilized result. The effectiveness and stability of the Two Gauss VMS method are further demonstrated through two numerical examples. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 361–375, 2014