Ricardo Oyarzúa

University of Bío-Bío | UBB · Department of Mathematics

In this paper, we propose a mass conservative pseudostress-based finite element method for solving the Stokes problem with both Dirichlet and mixed boundary conditions. We decompose the velocity by means of a Helmholtz decomposition and derive a three-field mixed variational formulation, where the pseudostress, the velocity, both in H(div), and an...

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that th...

In this paper we complement the study of a new mixed finite element scheme, allowing conservation of momentum and thermal energy, for the Boussinesq model describing natural convection and derive a reliable and efficient residual-based a posteriori error estimator for the corresponding Galerkin scheme in two and three dimensions. More precisely, by...

In this paper we consider a partially augmented fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop an a posteriori error analysis for the 2D and 3D versions of the associated mixed finite element scheme. Indeed, we derive two reliabl...

In this work we present and analyze a finite element scheme yielding discontinuous Galerkin approximations to the solutions of the stationary Boussinesq system for the simulation of non-isothermal flow phenomena. The model consists of a Navier–Stokes-type system, describing the velocity and the pressure of the fluid, coupled to an advection-diffusi...

We propose and analyze an unfitted method for a dual-dual mixed formulation of a class of Stokes models with variable viscosity depending on the velocity gradient, in which the pseudoestress, the velocity and its gradient are the main unknowns. On a fluid domain Ω with curved boundary Γ we consider a Dirichlet boundary condition and employ an appro...

We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables a...

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and sui...

In this paper, we propose and analyze a new momentum conservative mixed finite element method for the Navier–Stokes problem posed in nonstandard Banach spaces. Our approach is based on the introduction of a pseudostress tensor relating the velocity gradient with the convective term, leading to a mixed formulation where the aforementioned pseudostre...

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier--Stokes and Darcy--Forchheimer equations, and derive, though in a non-standard sense, a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method. For the reliability esti...

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-fiel...

In this paper we propose and analyze a fully-mixed finite element method for the steady-state model of fluidized beds. This numerical technique, which arises from the use of a dual-mixed approach in each phase, is motivated by a methodology previously applied to the stationary Navier–Stokes equations and related models. More precisely, we modify th...

In this work we present a new mixed finite element method for a class of steady-state natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the...

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and...

In this work we propose and analyze a new fully divergence-conforming finite element method for the numerical simulation of the Boussinesq problem, describing the motion of a non-isothermal incompressible fluid subject to a heat source. We consider the standard velocity-pressure formulation for the fluid flow equation and the dual-mixed one for the...

In this work we introduce and analyze a new augmented fully-mixed formulation for the stationary Navier–Stokes/Darcy coupled problem. Our approach employs, on the free-fluid region, a technique previously applied to the stationary Navier–Stokes equations, which consists of the introduction of a modified pseudostress tensor involving the diffusive a...

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...

In this paper we propose a new mixed-primal formulation for heat-driven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixed-point theorem and the generalized theory of saddle-point prob...

We propose and analyze a high order unfitted mixed finite element method for the pseudostress-velocity formulation of the Stokes problem with Dirichlet boundary condition on a fluid domain Ω with curved boundary Γ. The method consists of approximating Ω by a polyhedral subdomain D_h, with boundary Γ_h, where a Galerkin method is applied to approxim...

In this work we present and analyse a mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier--Stokes and the Darcy--Forchheimer equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers--Joseph--Saf...

We introduce a hybrid 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 f...

We propose and analyze a high order mixed finite element method for diffusion problems with Dirichlet boundary condition on a domain Ω with curved boundary Γ. The method is based on approximating Ω by a polygonal subdomain Dh, with boundary Γh, where a high order conforming Galerkin method is considered to compute the solution. To approximate the D...

We have recently proposed a new finite element method for a more general Boussinesq model in 2D given by the case in which the viscosity of the fluid depends on its temperature. Our approach is based on a pseudostress–velocity–vorticity mixed formulation for the momentum equations, which is suitably augmented with Galerkin-type terms, coupled with...

In this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully-mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach incorporates as additional unknowns a modified pseudostress tensor field and...

In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual‐based a poste...

In this paper we focus on the analysis of a mixed finite element method for a class of natural convection problems in two dimensions. More precisely, we consider a system based on the coupling of the steady-state equations of momentum (Navier–Stokes) and thermal energy by means of the Boussinesq approximation (coined the Boussinesq problem), where...

In this article, we analyse an augmented mixed finite element method for the steady Navier.Stokes equations. More precisely, we extend the recent results from Cama.no et al.. (2017, Analysis of an augmented mixed-FEM for the Navier.Stokes problem. Math. Comput., 86, 589.615) to the case of mixed no-slip and traction boundary conditions in different...

We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equa- tion suggested by the Oldroyd model for viscoelastic ow, coupled with the heat equation through a temperature-dependent viscosity of the uid and a convective term. T...

We introduce and analyze an augmented mixed finite element method for the Navier-Stokes-Brinkman problem with nonsolenoidal velocity. We employ a technique previously applied to the stationary Navier-Stokes equation, which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the...

In this paper we analyze the numerical approximation of a saddle-point problem posed in non-standard Banach spaces H(divp, Ω) × Lq(Ω), where H(divp, Ω):= {τ ϵ [L²(Ω)]ⁿ: div τ ϵ Lp(Ω)}, with p > 1 and q ϵ ℝ being the conjugate exponent of p and Ω ⊆ ℝⁿ (n ϵ {2, 3}) a bounded domain with Lipschitz boundary Λ. In particular, we are interested in derivi...

A new stress-based mixed variational formulation for the stationary Navier-Stokes equations with constant density and variable viscosity depending on the magnitude of the strain tensor, is proposed and analyzed in this work. Our approach is a natural extension of a technique applied in a recent paper by some of the authors to the same boundary valu...

In this paper we analyze a conforming finite element method for the numerical simulation of non-isothermal incompressible fluid flows subject to a heat source modelled by a generalized Boussinesq problem with temperature-dependent parameters. We consider the standard velocity-pressure formulation for the fluid flow equations which is coupled with a...

In an earlier work of us, a new mixed finite element scheme was developed for the Boussinesq model describing natural convection. Our methodology consisted of a fixed-point strategy for the variational problem that resulted after introducing a modified pseudostress tensor and the normal component of the temperature gradient as auxiliary unknowns in...

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier–Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law....

In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the coupling of the Navier–Stokes equations (with nonlinear viscosity) and the linear Darcy model, and derive a reliable and efficient residual-based a posteriori error estimator for the associated mixed finite element scheme. The finite e...

We propose and analyse an augmented mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by a class of nonlinear Navier–Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We...

In this paper we propose and analyze a new fully-mixed finite element method for the stationary Boussinesq problem. More precisely, we reformulate a previous primal-mixed scheme for the respective model by holding the same modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations f...

A new mixed variational formulation for the Na vier-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...

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 contin...

In this paper, we report on the main results concerning the solvability analysis of two new mixed variational formulations for the stationary Boussinesq problem. More precisely, we introduce mixed-primal and fully-mixed approaches, both of them suitably augmented with Galerkin-type equations, and show that the resulting schemes can be rewritten, eq...

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 "nonlinear-pseudostress" 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 mai...

This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška–Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that Raviart– Thomas elements of order...

In this article, we propose and analyze a new mixed variational formulation for the stationary Boussinesq problem. Our method, which uses a technique previously applied to the Navier–Stokes equations, is based first on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. Ne...

We propose and analyse a mixed finite element method for the nonstandard pseudostress-velocity formulation of the Stokes problem with varying density rho in R-d, d is an element of {2, 3}. Since the resulting variational formulation does not have the standard dual-mixed structure, we reformulate the continuous problem as an equivalent fixed-point p...

We propose and analyse a mixed finite element method with exactly divergence-free velocities for the numerical simulation of a generalized Boussinesq problem, describing the motion of a nonisothermal incompressible fluid subject to a heat source. The method is based on using divergence-conforming elements of order k for the velocities, discontinuou...

In this paper we introduce and analyze an augmented mixed finite element method for the coupling of quasi-Newtonian fluids and porous media. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–...

In this paper we develop an a posteriori error analysis of a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, the transmission conditions are given by the equilibrium of forces and the equality of the corresponding...

In this paper we develop the a priori and a posteriori error analyses of a mixed finite el-ement method for the coupling of fluid flow with nonlinear porous media flow. Flows are governed by the Stokes and nonlinear Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces,...

In this paper we analyze fully-mixed finite element methods for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The fully-mixed conce...

In this paper we analyze the well-posedness (unique solvability, stability, and Céa's esti-mate) of a family of Galerkin schemes for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal fo...

In this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Jo...

We consider a porous medium entirely enclosed within a fluid region and present a well-posed conform-ing mixed finite-element method for the corresponding coupled problem. The interface conditions refer to mass conservation, balance of normal forces and the Beavers–Joseph–Saffman law, which yields the introduction of the trace of the porous medium...