Brezzi Theory Research Articles

A note on the generalized Babuška–Brezzi theory: Revisiting the proof of the associated Strang error estimates

In this note we simplify the derivation of the error estimates for the generalized Babuška–Brezzi theory with Galerkin schemes defined in terms of approximate bilinear forms and functionals. More precisely, we provide a straight proof that makes no use of any translated continuous or discrete kernel nor of the distance between them, but of suitable upper bounds of the distances of each component of the Galerkin solution to any other member of the respective finite element subspace. In this way, the Strang error estimates are obtained simply by applying the aforementioned bounds along with the triangle inequality, so that they become cleaner and with fully explicit constants. The case in which the discrete bilinear forms can be evaluated at the continuous solution is also considered, which yields the consistency terms to appear separately from the distances to the subspaces, thus allowing the former to be handled independently from the latter.

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.

New non-augmented mixed finite element methods for the Navier–Stokes–Brinkman equations using Banach spaces

Abstract In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods.

New mixed finite element methods for the coupled Stokes and Poisson–Nernst–Planck equations in Banach spaces

In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwardsviaa postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuška–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Nečas–Babuška theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme.A priorierror estimates are derived, and examples of discrete spaces that fit the theory, include,e.g., Raviart–Thomas elements of orderkalong with piecewise polynomials of degree ≤k. In addition, the latter yield approximate local conservation of momentum for all three equations involved. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the aforementioned balance-preserving properties and the applicability of the proposed family of methods.

An Lp spaces-based mixed virtual element method for the two-dimensional Navier–Stokes equations

In this paper we extend the utilization of the Banach spaces-based formulations, usually employed for solving diverse nonlinear problems in continuum mechanics via primal and mixed finite element methods, to the virtual element method (VEM) framework and its respective applications. More precisely, we propose and analyze an [Formula: see text] spaces-based mixed virtual element method for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions. To this end, a dual-mixed approach determined by the introduction of a nonlinear tensor linking the usual pseudostress for the Stokes equations with the convective term is employed. As a consequence, this new tensor, say [Formula: see text], and the velocity [Formula: see text] of the fluid constitute the unknowns of the formulation, whereas the pressure is computed via a post-processing formula. The simplicity of the resulting VEM scheme is reflected by the absence of augmented terms, on the contrary to previous works on this and related models, and by the incorporation in it of only the projector onto the piecewise polynomial tensors and the usual stabilizer depending on the degrees of freedom of the virtual element subspace approximating [Formula: see text]. In turn, the non-virtual but explicit subspace given by the piecewise polynomial vectors of degree [Formula: see text] is employed to approximate [Formula: see text]. The corresponding solvability analysis 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. A Strang-type lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of [Formula: see text], the post-processed pressure, and a second post-processed approximation of [Formula: see text]. 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 Lpspaces-based formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations

AbstractIn this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the analysis, we need to make convenient choices of the Lebesgue and ${\textrm {H}}(div)$-type spaces to which the unknowns and test functions belong. The resulting coupled formulation is then written equivalently as a fixed-point operator, so that the classical Banach theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions and the aforementioned Neumann problem, are employed to establish the unique solvability of the continuous formulation. Under standard hypotheses satisfied by generic finite element subspaces, the associated Galerkin scheme is analysed similarly and the Brouwer theorem yields existence of a solution. The respective a priori error analysis is also derived. Then, Raviart–Thomas elements of order $k\ge 0$ for the pseudoheat and the velocity and discontinuous piecewise polynomials of degree $\le k$ for the pressure and the temperature are shown to satisfy those hypotheses in the two-dimensional case. Several numerical examples illustrating the performance and convergence of the method are reported, including an application into the equivalent problem of miscible displacement in porous media.

Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods

In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters. As a result, by taking appropriate limit of the stabilization parameters, we show that the HDG method converges to a primal conforming method and the WG method converge to a mixed conforming method.

Banach spaces-based analysis of a fully-mixed finite element method for the steady-state model of fluidized beds

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 the stress tensors of the fluid and solid phases by defining pseudostresses as phasic stresses that include shear, pressure, and convective effects. Next, we eliminate the pressures from the equations and derive constitutive relations depending only on the aforementioned pseudostresses and the velocities of the fluid and the particles. In this way, these variables, together with the skew-symmetric parts of the velocity gradients, also named vorticities, become the only unknowns of our variational formulation. As usual, the latter is obtained by testing against suitable functions, and then integrating and integrating by parts, respectively, the equilibrium and the constitutive equations. The particle pressure, a known function of the concentration, is given as a datum, and the fluid pressure is computed afterwards via a postprocessing formula. The continuous setting, lying in a Banach spaces framework rather than in a Hilbertian one, is rewritten as an equivalent fixed-point equation, and hence the well-posedness analysis is carried out by combining the Babusˇka–Brezzi theory, the Banach–Necˇas–Babusˇka Theorem, and the classical Banach fixed-point Theorem. Thus, existence of a unique solution in a closed ball is guaranteed for sufficiently small data. In turn, the associated Galerkin scheme is introduced and analyzed analogously, so that, under suitable assumptions on generic finite element subspaces, and for sufficiently small data as well, the Brouwer and Banach fixed-point Theorems allow to conclude existence and uniqueness of solution, respectively. Specific finite element subspaces satisfying the required hypotheses are described, and optimal a priori error estimates are derived. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence, are reported.

Well-posedness of a family of degenerate parabolic mixed equations

In this study, we present an abstract framework for analyzing a family of linear degenerate parabolic mixed equations. We combine the theory of degenerate parabolic equations with the classical Babuška–Brezzi theory for linear mixed stationary equations to deduce sufficient conditions to prove the well-posedness of the problem. Finally, we illustrate the application of the abstract framework based on examples from physical science applications, including fluid dynamics models and electromagnetic problems.

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 priori and a posteriori error analyses of a high order unfitted mixed-FEM for Stokes flow

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 Dh, with boundary Γh, where a Galerkin method is applied to approximate the solution, and on a transferring technique, based on integrating the extrapolated discrete gradient of the velocity, to approximate the Dirichlet boundary data on the computational boundary Γh. The associated Galerkin scheme is defined by Raviart–Thomas elements of order k≥0 for the pseudostress and discontinuous polynomials of degree k for the velocity. Provided suitable hypotheses on the mesh near the boundary Γ, we prove well-posedness of the Galerkin scheme by means of the Babuška–Brezzi theory and establish the corresponding optimal convergence of O(hk+1). Moreover, for the case when Γh is constructed through a piecewise linear interpolation of Γ, we propose a reliable and quasi-efficient residual-based a posteriori error estimator. Its definition makes use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough. Numerical experiments illustrate the performance of the scheme, show the behavior of the associated adaptive algorithm and validate the theory.

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity, and pressure with non-constant viscosity. The analysis is performed by the classical Babuška–Brezzi theory, and we state that any inf–sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates, which are further confirmed through computational examples.

Mixed displacement–rotation–pressure formulations for linear elasticity

We propose a new locking-free family of mixed finite element and finite volume element methods for the approximation of linear elastostatics, formulated in terms of displacement, rotation vector, and pressure. The unique solvability of the three-field continuous formulation, as well as the well-definiteness and stability of the proposed Galerkin and Petrov–Galerkin methods, is established thanks to the Babuška–Brezzi theory. Optimal a priori error estimates are derived using norms robust with respect to the Lamé constants, turning these numerical methods particularly appealing for nearly incompressible materials. We exemplify the accuracy (in a suitably weighted norm), as well the applicability of the new formulation and the mixed schemes by conducting a number of computational tests in 2D and 3D, also including cases not covered by our theoretical analysis.

A mixed virtual element method for a pseudostress-based formulation of linear elasticity

In this paper we introduce and analyze a mixed virtual element method (mixed-VEM) for a pseudostress-displacement formulation of the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. We follow a previous work by some of the authors, and employ a mixed formulation that does not require symmetric tensor spaces in the finite element discretization. More precisely, the main unknowns here are given by the pseudostress and the displacement, whereas other physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through simple postprocessing formulae in terms of the pseudostress variable. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, we utilize a well-known local projector onto a suitable polynomial subspace to define a calculable version of our discrete bilinear form, whose continuous version requires information of the variables on the interior of each element. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Babuška–Brezzi theory. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solutions as well as for the fully computable projections of them. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. In addition, this postprocessing formula can also be applied to the postprocessed stress tensor. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

A minimax approach for the study of systems of variational equations and related Galerkin schemes

The aim of this work is to analyze the existence of a solution for a quite general variational inequalities system, which includes those appearing in the theory of mixed variational equations, the so-called Babuška–Brezzi theory . It is done by means of a minimax inequality, which also provides us with a estimate of the norm of the solution. Then, we derive a numerical method of the Galerkin type to approximate the solution of such a system. Its stability follows from the mentioned control of the norm of the solution.

Mixed Methods for a Stream-Function – Vorticity Formulation of the Axisymmetric Brinkman Equations

This paper is devoted to the numerical analysis of a family of finite element approximations for the axisymmetric, meridian Brinkman equations written in terms of the stream-function and vorticity. A mixed formulation is introduced involving appropriate weighted Sobolev spaces, where well-posedness is derived by means of the Babuska---Brezzi theory. We introduce a suitable Galerkin discretization based on continuous piecewise polynomials of degree $$k\ge 1$$kź1 for all the unknowns, where its solvability is established using the same framework as the continuous problem. Optimal a priori error estimates are derived, which are robust with respect to the fluid viscosity, and valid also in the pure Darcy limit. A few numerical examples are presented to illustrate the convergence and performance of the proposed schemes.

A vorticity-based fully-mixed formulation for the 3D Brinkman–Darcy problem

We propose and analyze a fully-mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium (an array of low concentration fixed particles), described by Brinkman equations, and its interaction with non-viscous flow within classical porous media governed by Darcy’s law. The system is formulated in terms of velocity and pressure in the porous medium, together with vorticity, velocity and pressure of the viscous fluid. In addition, and for sake of the analysis, the tangential component of the vorticity is supposed to vanish on the whole boundary of the Brinkman domain, whereas null normal components of both velocities are assumed on the respective boundaries, except on the interface where suitable transmission conditions are considered. In this way, the derivation of the corresponding mixed variational formulation leads to a Lagrange multiplier enforcing the pressure continuity across the interface, whereas mass balance results from essential boundary conditions on each domain. As a consequence, a typical saddle-point operator equation is obtained, and hence the classical Babuška–Brezzi theory is applied to establish the well-posedness of the continuous and discrete schemes. In particular, we remark that the continuous and discrete inf–sup conditions of the main bilinear form are proved by using suitably chosen injective operators to get lower bounds of the corresponding suprema, which constitutes a previously known technique, recently denominated T-coercivity. In turn, and consistently with the above, the stability of the Galerkin scheme requires that the curl of the finite element subspace approximating the vorticity be contained in the space where the discrete velocity of the fluid lives, which yields Raviart–Thomas and Nédélec finite element subspaces as feasible choices. Then we show that the aforementioned constraint can be avoided by augmenting the mixed formulation with a residual arising from the Brinkman momentum equation. Finally, several numerical examples illustrating the satisfactory performance of the methods and confirming the theoretical rates of convergence are reported.

Stable finite element methods preserving $$\nabla \cdot \varvec{B}=0$$ ∇ · B = 0 exactly for MHD models

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss's law, namely $$\nabla \cdot \varvec{B}=0$$ź·B=0. In contrast to most existing approaches that eliminate the electrical field variable $$\varvec{E}$$E and give a direct discretization of the magnetic field, our new approach discretizes the electric field $$\varvec{E}$$E by Nedelec type edge elements for $$H(\mathrm {curl})$$H(curl), while the magnetic field $$\varvec{B}$$B by Raviart---Thomas type face elements for $$H(\mathrm {div})$$H(div). As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for the Picard linearization of the fully nonlinear systems by using the Brezzi theory. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

A priori and a posteriori error analyses of a pseudostress-based mixed formulation for linear elasticity

In this paper we present the a priori and a posteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. More precisely, the approach introduced here is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. In addition, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second element-by-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order k≥0 for the pseudostress and piecewise polynomials of degree ≤k for the displacement can be utilized. Moreover, we remark that in the 3D case the number of unknowns behaves approximately as 9 times the number of elements (tetrahedra) of the triangulation when k=0. This factor increases to 12.5 when one uses the classical PEERS. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the mixed finite element scheme. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the associated adaptive algorithm, are provided.

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

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 Babuska---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 $$k \ge 0$$kź0 for the approximation of the velocity field, piecewise continuous polynomials of degree $$k+1$$k+1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi---Douglas---Marini elements of order $$k+1$$k+1 for the approximation of velocity, piecewise continuous polynomials of degree $$k+2$$k+2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual-based a posteriori error estimator. Finally, we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes. Additional examples of cases not covered by our theory are also presented.