An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

Abstract

a b s t r a c tThis paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi-Douglas-Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart-Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška-Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

“…Here, we propose a new class of stabilized finite element approximations of the Brinkman equations, written in terms of the velocity, vorticity, and pressure fields. One of the main goals of the present approach is to build different families of finite elements to approximate the model problem with the liberty of choosing any combination of the finite element subspaces of the continuous spaces and extend recent results given in [24], where a new stabilized finite element approximation for the Stokes equations was analyzed using an extension of the Babuška-Brezzi theory (cf. [31,32]).…”

“…However, up to our knowledge, the Brinkman problem has been considered using mixed vorticity-velocitypressure formulations only very recently [19], in where a dual-mixed formulation has been analyzed at the continuous and discrete levels using the Babuška-Brezzi theory and optimal error estimates are provided.The so-called augmented mixed finite elements (also known as Galerkin least-squares methods [12,20,21]) can be regarded as a stabilization technique where some terms are added to the variational formulation so that, either the resulting augmented variational formulations are defined by strongly coercive bilinear forms (see, e.g., [22]), or they enable to bypass the kernel property, which is very difficult to obtain in practice, or they allow the fulfillment of the inf-sup condition at the continuous and discrete levels in mixed formulations ([23]). This approach has been considered in, for example, [5,[24][25][26][27][28][29] for Stokes, generalized Stokes (in velocity-pseudostress formulation), coupling of quasi-Newtonian fluids and porous media, and Navier-Stokes equations, and in [30] for an augmented mixed formulation applied to elliptic problems with mixed boundary conditions.Here, we propose a new class of stabilized finite element approximations of the Brinkman equations, written in terms of the velocity, vorticity, and pressure fields. One of the main goals of the present approach is to build different families of finite elements to approximate the model problem with the liberty of choosing any combination of the finite element subspaces of the continuous spaces and extend recent results given in [24], where a new stabilized finite element approximation for the Stokes equations was analyzed using an extension of the Babuška-Brezzi theory (cf.…”

“…This approach has been considered in, for example, [5,[24][25][26][27][28][29] for Stokes, generalized Stokes (in velocity-pseudostress formulation), coupling of quasi-Newtonian fluids and porous media, and Navier-Stokes equations, and in [30] for an augmented mixed formulation applied to elliptic problems with mixed boundary conditions.Here, we propose a new class of stabilized finite element approximations of the Brinkman equations, written in terms of the velocity, vorticity, and pressure fields. One of the main goals of the present approach is to build different families of finite elements to approximate the model problem with the liberty of choosing any combination of the finite element subspaces of the continuous spaces and extend recent results given in [24], where a new stabilized finite element approximation for the Stokes equations was analyzed using an extension of the Babuška-Brezzi theory (cf. [31,32]).…”

“…The so-called augmented mixed finite elements (also known as Galerkin least-squares methods [12,20,21]) can be regarded as a stabilization technique where some terms are added to the variational formulation so that, either the resulting augmented variational formulations are defined by strongly coercive bilinear forms (see, e.g., [22]), or they enable to bypass the kernel property, which is very difficult to obtain in practice, or they allow the fulfillment of the inf-sup condition at the continuous and discrete levels in mixed formulations ([23]). This approach has been considered in, for example, [5,[24][25][26][27][28][29] for Stokes, generalized Stokes (in velocity-pseudostress formulation), coupling of quasi-Newtonian fluids and porous media, and Navier-Stokes equations, and in [30] for an augmented mixed formulation applied to elliptic problems with mixed boundary conditions.…”

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

“…Several numerical methods exploit these properties, as for instance, different formulations based on least-squares, stabilization techniques, mixed finite elements, spectral discretizations, and hybridizable discontinuous Galerkin methods (see for instance [3,4,8,12,14,18,19,[21][22][23]26,[34][35][36], and the references therein). For the generalized Stokes problem written in velocity-vorticity-pressure variables, we mention [6] where an augmented mixed formulation based on RT k − P k+1 − P k+1 (with continuous pressure approximation) finite elements has been developed and analyzed.…”