Pseudostress Tensor Research Articles

A Banach spaces-based fully mixed virtual element method for the stationary two-dimensional Boussinesq equations

In this paper we extend recent results obtained for the Navier–Stokes equations to propose and analyze a new fully mixed virtual element method (mixed-VEM) for the stationary two-dimensional Boussinesq equations appearing in non-isothermal flow phenomena. The model consists of a Navier–Stokes type system, modeling the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The variational formulation is based on the introduction of the additional unknowns given by a modified pseudostress tensor, which depends on the pressure, and the diffusive and convective terms of the fluid, and the pseudoheat vector, which involves the temperature, its gradient, and the velocity. As a consequence of the former, the pressure is eliminated from the system, but computed afterwards via a post-processing formula. In turn, for the Galerkin approximation we follow the approach employed in a previous work introducing for the first time an Lp spaces-based mixed-VEM for the Navier–Stokes equations, and couple it with a similar mixed-VEM for the convection–diffusion equation modeling the temperature. The solvability analysis of the resulting coupled discrete scheme is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach-Nečas-Babuška theorem, both in subspaces of Banach spaces. The first Strang lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of the pseudostress tensor, the pseudoheat vector, and the post-processed pressure. Finally, several numerical results, illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory, are reported.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

Coupled mixed finite element and finite volume methods for a solid velocity-based model of multidimensional sedimentation

In this paper we introduce and analyze a model of sedimentation based on a solid velocity formulation. A particular feature of the governing equations is given by the fact that the velocity field is non-divergence free. We introduce extra variables such as the pseudostress tensor relating the velocity gradient with the pressure, thus leading to a mixed variational formulation consisting of two systems of equations coupled through their source terms. A result of existence and uniqueness of solutions is shown by means of a fixed-point strategy and the help of the Babuška–Brezzi theory and Banach theorem. Additionally, we employ suitable finite dimensional subspaces to approximate both systems of equations via associated mixed finite element methods. The well-posedness of the resulting coupled scheme is also treated via a fixed-point approach, and hence the discrete version of the existence and uniqueness result is derived analogously to the continuous case. The above is then combined with a finite volume method for the transport equation. Finally, several numerical results illustrating the performance of the proposed model and the full numerical scheme, and confirming the theoretical rates of convergence, are presented.

A five-field mixed formulation for stationary magnetohydrodynamic flows in porous media

We introduce and analyze a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman–Forchheimer equations coupled with the Maxwell equations. Besides the velocity, magnetic field and a Lagrange multiplier associated to the divergence-free condition of the magnetic field, a convenient translation of the velocity gradient and the pseudostress tensor are introduced as further unknowns. As a consequence, we obtain a five-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. The resulting mixed scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and a sufficiently small data assumption, are applied to prove the unique solvability of the continuous and discrete systems. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. The finite element discretization involves Raviart–Thomas elements of order k≥0 for the pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity and the translation of the velocity gradient, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k+1 for the associated Lagrange multiplier. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

Analysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics

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 the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided.

Displacement‐pseudostress formulation for the linear elasticity spectral problem

AbstractIn this paper we analyze a mixed displacement‐pseudostress formulation for the elasticity eigenvalue problem. We propose a finite element method to approximate the pseudostress tensor with Raviart–Thomas elements and the displacement with piecewise polynomials. With the aid of the classic theory for compact operators, we prove that our method is convergent and does not introduce spurious modes. Error estimates for the proposed method are derived. Finally, we report some numerical tests supporting the theoretical results.

A three-field Banach spaces-based mixed formulation for the unsteady Brinkman–Forchheimer equations

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 are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k≥0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media

In this paper we study a stationary double-diffusive natural convection problem in porous media given by a Navier-Stokes/Brinkman type system, for describing the velocity and the pressure, coupled to a vector advection-diffusion equation relate to the heat and substance concentration, of a viscous fluid in a porous media with physical boundary conditions. The model problem is rewritten in terms of a first-order system, without the pressure, based on the introduction of the strain tensor and a nonlinear pseudo-stress tensor in the fluid equations. After a variational approach, the resulting weak model is then augmented using appropriate redundant penalization terms for the fluid equations along with a standard primal formulation for the heat and substance concentration. Then, it is rewritten as an equivalent fixed-point problem. Well-posedness results for both the continuous and the discrete schemes are stated, as well as the respective convergence result under certain regularity assumptions combined with the Lax-Milgram theorem, and the Banach and Brouwer fixed-point theorems. In particular, Raviart-Thomas elements of order k are used for approximating the pseudo-stress tensor, piecewise polynomials of degree ≤k and ≤k+1 are utilized for approximating the strain tensor and the velocity, respectively, and the heat and substance concentration are approximated by means of Lagrange finite elements of order ≤k+1. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed semi-augmented mixed-primal scheme.

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman–Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška–Brezzi’s theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimala priorierror estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.

Analysis of a momentum conservative mixed‐FEM for the stationary Navier–Stokes problem

AbstractIn 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 pseudostress tensor and the velocity are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree for the pseudostress tensor and discontinuous piece–wise polynomial elements of degree for the velocity. With this choice of spaces, the equilibrium equation is exactly satisfied if the external force belongs to the velocity discrete space, thus the method conserves momentum, which constitutes one of the main feature of our approach. For both, the continuous and discrete problems, the Banach–Nečas–Babuška and Banach's fixed‐point theorems are employed to prove unique solvability. We also provide the convergence analysis and particularly prove that the error decay with optimal rate of convergence. 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. Finally, several numerical results illustrating the performance of the method are provided.

A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem

In this paper we analyze a virtual element method (VEM) for a pseudostress formulation of the Stokes eigenvalue problem. This formulation allows to eliminate the velocity and the pressure, leading to an elliptic formulation where the only unknown is the pseudostress tensor. Adapting the non-compact operator theory, we prove that our method provides a correct approximation of the spectrum and is spurious free. We derive optimal a priori error estimates, which we confirm with some numerical tests. Also we present a computational spurious analysis of the proposed method.

A fully‐mixed finite element method for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

AbstractIn this work we present and analyze a fully‐mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non‐Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well‐known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported.

A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

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 the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in $$\mathrm {L}^4$$ . As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable $$\mathrm {H}(\mathrm {div})$$ -type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Necas–Babuska theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order $$k\ge 0$$ for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree $$\le k + 1$$ are employed for the velocity, and piecewise polynomials of degree $$\le k$$ for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy

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 Navier–Stokes equations for the fluid and a vector unknown involving the temperature, its gradient and the velocity. The introduction of these further unknowns lead to a mixed formulation where the aforementioned pseudostress tensor and vector unknown, together with the velocity and the temperature, are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the pseudostress tensor and the vector unknown, and discontinuous piece-wise polynomial elements of degree k for the velocity and temperature. With this choice of spaces, both, momentum and thermal energy, are conserved if the external forces belong to the velocity and temperature discrete spaces, respectively, which constitutes one of the main feature of our approach. We prove unique solvability for both, the continuous and discrete problems and provide the corresponding convergence analysis. Further variables of interest, such as the fluid pressure, the fluid vorticity, the fluid velocity gradient, and the heat-flux can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. Finally, several numerical results illustrating the performance of the method are provided.

A five-field augmented fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem

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 and convective terms, together with the pressure. In addition, by using the incompressibility condition, the pressure is eliminated, and since the convective term forces the free-fluid velocity to live in a smaller space than usual, we augment the resulting formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations. On the other hand, in the Darcy region we apply the usual dual-mixed formulation, which yields the introduction of the trace of the porous media pressure as an associated Lagrange multiplier. The latter is connected with the fact that one of the transmission conditions involving mass conservation becomes essential and must be imposed weakly. In this way, we obtain a five-field formulation where the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the aforementioned Lagrange multiplier, are the corresponding unknowns. The well-posedness analysis is carried out by combining the classical Babuška–Brezzi theory and the Banach fixed-point theorem. 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-posed and convergent. In particular, Raviart–Thomas elements of lowest order for the pseudostress and the Darcy velocity, continuous piecewise linear polynomials for the free-fluid velocity, piecewise constants for the Darcy pressure, together with continuous piecewise linear elements for the Lagrange multiplier, constitute feasible choices. Finally, we provide several numerical results illustrating the performance of the Galerkin method and confirming the theoretical rates of convergence.

A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

AbstractWe propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

A posteriori error analysis of a mixed virtual element method for a nonlinear Brinkman model of porous media flow

In this paper we present an a posteriori error analysis of a mixed-VEM discretization for a nonlinear Brinkman model of porous media flow, which has been proposed by the authors in a previous work. Therein, the system is formulated in terms of a pseudostress tensor and the velocity gradient, whereas the velocity and the pressure of the fluid are computed via postprocessing formulas. Furthermore, the well-posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established. We now propose reliable and efficient residual-based a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem. The resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes. For the analysis we make use of a global inf–sup condition, Helmholtz decomposition, local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions. Finally, we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm.

EFFECT OF ISOMALTULOSE CONSUMPTION DURING THE PREGNANCY IN NEW BORN RATS. (PRELIMINARY RESULTS)

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 an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.

Impaired placental vessel formation in mice lacking of adenosine A2A receptor

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 an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.

A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

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 an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.