Analysis of an augmented mixed-FEM for the Navier-Stokes problem

In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinearpseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach’s fixed point Theorem and Lax-Milgram’s Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea’s estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k + 1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.Comisión Nacional de Investigación Científica y TecnológicaMinistry of Education, Youth and Sports of the Czech Republi

A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

We introduce and analyze a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The model consists of an elastic body which is subject to a given incident wave that travels in the fluid surrounding it. Actually, the fluid is supposed to occupy an annular region, and hence a Robin boundary condition imitating the behavior of the scattered field at infinity is imposed on its exterior boundary, which is located far from the obstacle. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We first apply dual-mixed approaches in both domains, and then employ the governing equations to eliminate the displacement u of the solid and the pressure p of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the Cauchy stress tensor and the rotation of the solid, together with the gradient of p and the traces of u and p on the boundary of the fluid, constitute the unknowns of the coupled problem. Next, we show that suitable decompositions of the spaces to which the stress and the gradient of p belong, allow the application of the Babuška–Brezzi theory and the Fredholm alternative for analyzing the solvability of the resulting continuous formulation. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. Then, the analysis of the discrete method relies on a stable decomposition of the corresponding finite element spaces and also on a classical result on projection methods for Fredholm operators of index zero. Finally, some numerical results illustrating the theory are presented

An augmented mixed finite element method for the Navier-Stokes equations with variable viscosity

A new mixed variational formulation for the Navier–Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier-Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additional unknown, and the Dirichlet boundary condition. The resulting augmented scheme is then written equivalently as a fixed point equation, and hence the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. No discrete inf-sup conditions are required for the well-posedness of the Galerkin scheme, and hence arbitrary finite element subspaces of the respective continuous spaces can be utilized. In particular, given an integer k ≥ 0, piecewise polynomials of degree ≤ k for the gradient of velocity, Raviart-Thomas spaces of order k for the pseudostress, and continuous piecewise polynomials of degree ≤ k + 1 for the velocity, constitute feasible choices. Finally, optimal a priori error estimates are derived, and several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.Comisión Nacional de Investigación Científica y Tecnológica (Chile)Universidad del Bío-BíoMinistry of Education, Youth and Sports of the Czech Republi

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

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Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity

We introduce a numerical method for the approximation of linear poroelasticity equations, representing the interaction between the non-viscous filtration flow of a fluid and the linear mechanical response of a porous medium. In the proposed formulation, the primary variables in the system are the solid displacement, the fluid pressure, the fluid flux, and the total pressure. A discontinuous finite volume method is designed for the approximation of solid displacement using a dual mesh, whereas a mixed approach is employed to approximate fluid flux and the two pressures. We focus on the stationary case and the resulting discrete problem exhibits a double saddle-point structure. Its solvability and stability are established in terms of bounds (and of norms) that do not depend on the modulus of dilation of the solid. We derive optimal error estimates in suitable norms, for all field variables; and we exemplify the convergence and locking-free properties of this scheme through a series of numerical tests

Locking-free finite element methods for poroelasticity

We propose a new formulation along with a family of finite element schemes for the approximation of the interaction between fluid motion and linear mechanical response of a porous medium, known as Biot's consolidation problem. The steady-state version of the system is recast in terms of displacement, pressure, and volumetric stress, and both continuous and discrete formulations are analyzed as compact perturbations of invertible problems employing a Fredholm argument. In particular, the error estimates are derived independently of the Lamé constants. Numerical results indicate the satisfactory performance and competitive accuracy of the introduced methods

Error analysis of an augmented mixed method for the Navier–Stokes problem with mixed boundary conditions

In this paper we analyze an augmented mixed finite element method for the steady Navier-Stokes equations. More precisely, we extend the recent results from Camano˜ et al. (2017) to the case of mixed no-slip and traction) boundary conditions in different parts of the boundary, and introduce and analyze a new pseudostress-velocity augmented mixed formulation for the fluid flow problem. The well-posedness analysis is carried out by combining the classical Babuska-Brezzi theory and Banach’s fixed-point Theo- ˇ rem. A proper adaptation of the arguments exploited in the continuous analysis allows us to state suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme is well-defined. For instance, Raviart-Thomas elements of order k > 0 and continuous piecewise polynomials of degree k + 1 for the nonlinear pseudo-stress tensor and velocity, respectively, yield optimal convergence rates. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the proposed discretization. The proof of reliability hinges on the global inf–sup condition and the local approximation properties of the Clement interpolant, whereas the efficiency of the estimator follows from a inverse ´ inequalities and localization via edge–bubble functions. A set of numerical results exemplifies the performance of the augmented method with mixed boundary conditions. The tests also confirm the reliability and efficiency of the estimator, and show the performance of the associated adaptive algorithm

Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

We present an a priori and a posteriori error analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For the a priori error analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of order k ≥ 0 for the fluid flux, discontinuous polynomials of degree k for the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-based a posteriori error estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposed a posteriori error indicator