A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday

Abstract

This article is concerned with the Stokes system with nonhomogeneous source terms and nonhomogeneous Dirichlet boundary condition. First, we reformulate the problem in its dual mixed form, and then, we study its corresponding well-posedness. Next, in order to circumvent the well-known Babuška-Brezzi condition, we analyze a stabilized formulation of the resulting approach. Additionally, we endow the scheme with an a posteriori error estimator that is reliable and efficient. Finally, we provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice. KEYWORDSa posteriori error estimates, augmented mixed formulation, Ritz projection of the error INTRODUCTIONIn the work of Cai et al, 1 a dual mixed finite element method for the incompressible fluid flow was introduced and analyzed. The approach there follows the ideas developed in the other work of Cai et al, 2 ie, the incompressible fluid flow is reformulated using the new variable so-called pseudostress, which is in relation with the pressure and gradient of the velocity. The main advantage of this new variable is the accurate approximation to physical quantities such as the stress and vorticity, allowing to use the pair of conforming Raviart-Thomas with discontinuous polynomial as the finite element space. Furthermore, in order to obtain more flexibility in the finite element spaces, the stabilization of this approach has been studied in the work of Figueroa et al. 3 In addition, its corresponding extension to quasi Newtonian flows and Brinkman model were developed in the works of Gatica et al 4 and Barrios et al, 5 respectively. On the other hand, concerning linear elasticity problem, in the work of Barrios et al, 6 the authors present an alternative a posteriori error estimator to the previous one developed in a former work. 7 This approach is based on the Ritz projection Int J Numer Meth Fluids. 2020;92:509-527. wileyonlinelibrary.com/journal/fld

A note on a posteriori error analysis for dual mixed methods with mixed boundary conditions

“…There, a posteriori error estimator with seven terms per element is deduced, less than the thirteen terms needed in the local a posteriori error estimator previously obtained in [6], for the same finite element spaces. Additionally, this kind of a posteriori error estimator, at least, have been developed satisfactorily in different directions, for example, the Poisson problem is studied in [15], Darcy flow in [12] and [13], the Stokes system in [5] and [10], the Brinkman model in [11], linear elasticity in [7,8] and the Oseen equations in [14].…”

An A Posteriori Error Estimator for a Non Homogeneous Dirichlet Problem Considering a Dual Mixed Formulation

“…This is performed with the help of the Ritz projection of the error, and covers the reliability and efficiency of the estimator. It is important to remark that this technique has been successfully applied to other problems, such as the the Brinkman model in [2], the Darcy flow in [6] and [7], the Stokes system in [3] and [5], and the Oseen equations in [8], for example.…”