Abstract
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–Nečas–Babuška 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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
1 Introduction
The phenomenon of double-diffusive convection, in which two scalar fields, such as heat and concentration of a solute, affect the density distribution in a fluid-saturated porous medium, has a wide range of applications, including processes arising in chemical engineering, energy technology, geophysics, and oceanography. In particular, some applications include groundwater system in karst aquifers, chemical processing, convective flow of carbon nanotubes, propagation of biological fluids, and simulation of bacterial bioconvection and thermohaline circulation problems (see, e.g. [1, 3, 6, 21, 35] to name a few). In this regard, we remark that much of the research in porous medium has been focused on the use of Darcy’s law. However, this constitutive equation becomes unreliable to model the flow of fluids through highly porous media at higher Reynolds numbers, as in the above applications. To avoid this inconvenient, a first alternative is to employ the Brinkman model [5], which describes Stokes flows through array of obstacles, and therefore can be applied precisely to that kind of media. Another possible option is the Forchheimer law [22], which accounts for faster flows by including a nonlinear inertial term. According to the above, the Brinkman–Forchheimer equation (see, e.g. [13, 31]), which combines the advantages of both models, has been used for fast flows in highly porous media. Moreover, this fact has motivated the introduction of the corresponding coupling with a system of advection-diffusion equations (also called double-diffusion equations), through convective terms and the body force.
In this context, and up to the authors’ knowledge, one of the first works in analyzing the coupling of the incompressible Brinkman–Forchheimer and double-diffusion equations is [28]. In there, the authors propose a velocity–pressure–temperature-concentration variational formulation and discuss the corresponding analysis of existence, uniqueness, and regularity of solution. To that end, a Galerkin method was employed to prove that the problem has at least one solution and that, under a smallness data assumption, a uniqueness result is established. Later on, the global solvability of a time-dependent double-diffusive convection system coupled with a linearized version of the Brinkman–Forchheimer equations was introduced and analyzed in [30]. In particular, the authors prove that the global solvability in \(\mathrm {L}^2\)-spaces holds true for the 3-dimensional case. More recently, in [34] a finite volume method was adopted to solve the coupling of the unsteady Brinkman–Forchheimer and double-diffusion equations. The focus of this work was on the validity of the Brinkman–Forchheimer model when various combinations of the thermal Rayleigh number, inclination angle, permeability ratio, thermal conductivity and buoyancy ratio are considered. Meanwhile, a \({\mathbf {H}}(\mathrm {div})\)-conforming method for double-diffusion equations but coupled with the stationary Navier–Stokes–Brinkman model was analyzed in [6]. Here, the solvability analysis results as a combination of compactness arguments and fixed-point theory. The corresponding numerical scheme is based on Brezzi–Douglas–Marini (BDM) elements of order k for the velocity, discontinuous elements of order \(k-1\) for the pressure, and Lagrangian finite elements of order k for temperature and the concentration of a solute. We observe that this formulation produces exactly divergence-free velocity approximations.
According to the above bibliographic discussion, the goal of the present paper is to develop and analyze a new fully-mixed formulation for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations and study its numerical approximation by a mixed finite element method. To that end, unlike previous works, we introduce the pseudostress tensor as in [11] and subsequently eliminate the pressure unknown using the incompressibility condition. In turn, and in order to enforce conservation of momentum in a physically compatible way, we proceed similarly to [7, 9] and introduce the pseudoheat and pseudodiffusive vectors as additional unknowns. Furthermore, the difficulty given by the fact that the fluid velocity lives in \(\mathrm {H}^1\) instead of \(\mathrm {L}^2\) as usual, is resolved as in [11, 25] by augmenting the variational formulation with residuals arising from the constitutive equation and the Dirichlet boundary condition on the velocity, which forces both the temperature and concentration fields to live in \(\mathrm {L}^4\), and consequently the pseudoheat and pseudodiffusive vectors in a suitable \(\mathrm {H}(\mathrm {div})\)-type Banach space. Then, following [17, 25] and [9], we combine classical fixed-point arguments with the Lax–Milgram and Banach–Nečas–Babuška theorems to prove the well-posedness of both the continuous and discrete formulations. In particular, for the continuous formulation, and under a smallness data assumption, we prove existence and uniqueness of solution by means of a fixed-point strategy where the Schauder (for existence) and Banach (for uniqueness) fixed-point theorems are employed. Using similar arguments (but applying Brower’s fixed-point theorem instead of Schauder’s for the existence result) we prove the well-posedness of the discrete problem for arbitrary conforming discrete spaces. In addition, applying an ad-hoc Strang-type lemma in Banach spaces, we are able to derive the corresponding a priori error estimates. Next, employing Raviart–Thomas spaces of order \(k\ge 0\) for approximating the pseudostress tensor, the pseudoheat and pseudodiffusive vectors, continuous piecewise polynomials of degree \(k + 1\) for velocity, and piecewise polynomials of degree k for the temperature and concentration fields, we prove that the method is convergent with optimal rate.
The rest of this work is organized as follows. The remainder of this section describes standard notation and functional spaces to be employed throughout the paper. In Sect. 2 we introduce the model problem and derive its augmented fully-mixed variational formulation. Next, in Sect. 3 we establish the well-posedness of this continuous scheme by means of a fixed-point strategy and Schauder and Banach fixed-point theorems. The corresponding Galerkin system is introduced and analyzed in Sect. 4, where the discrete analogue of the theory used in the continuous case is employed to prove existence and uniqueness of solution. In Sect. 5, an ad-hoc Strang-type lemma in Banach spaces is utilized to derive the corresponding a priori error estimate and the consequent rates of convergence. Finally, in Sect. 6 we report some numerical experiments illustrating the accuracy and flexibility of our augmented fully-mixed finite element method.
1.1 Preliminary Notations
Let \(\Omega \subset \mathrm {R}^n, n\in \{2,3\}\), be a bounded domain with polyhedral boundary \(\Gamma \), and let \({{\mathbf {n}}}\) be the outward unit normal vector on \(\Gamma \). Standard notation will be adopted for Lebesgue spaces \(\mathrm {L}^p(\Omega )\) and Sobolev spaces \(\mathrm {W}^{s,p}(\Omega )\), with \(s\in \mathrm {R}\) and \(p > 1\), whose corresponding norms, either for the scalar, vectorial, or tensorial case, are denoted by \(\Vert \cdot \Vert _{0,p;\Omega }\) and \(\Vert \cdot \Vert _{s,p;\Omega }\), respectively. In particular, given a non-negative integer m, \(\mathrm {W}^{m,2}(\Omega )\) is also denoted by \(\mathrm {H}^m(\Omega )\), and the notations of its norm and seminorm are simplified to \(\Vert \cdot \Vert _{m,\Omega }\) and \(|\cdot |_{m,\Omega }\), respectively. By \({\mathbf {M}}\) and \({\mathbb {M}}\) we will denote the corresponding vectorial and tensorial counterparts of the generic scalar functional space \(\mathrm {M}\), and \(\Vert \cdot \Vert \), with no subscripts, will stand for the natural norm of either an element or an operator in any product functional space. In turn, for any vector field \({{\mathbf {v}}}=(v_i)_{i=1,n}\), we let \(\nabla \,{{\mathbf {v}}}\) and \(\mathrm {div}({{\mathbf {v}}})\) be its gradient and divergence, respectively. Furthermore, for any tensor fields \({\varvec{\tau }}=(\tau _{ij})_{i,j=1,n}\) and \({\varvec{\zeta }}=(\zeta _{ij})_{i,j=1,n}\), we let \(\mathbf {div}({\varvec{\tau }})\) be the divergence operator \(\mathrm {div}\) acting along the rows of \({\varvec{\tau }}\), and define the transpose, the trace, the tensor inner product, and the deviatoric tensor, respectively, as
where \({\mathbb {I}}\) is the identity matrix in \(\mathrm {R}^{n\times n}\). In what follows, when no confusion arises, \(|\cdot |\) will denote the Euclidean norm in \(\mathrm {R}^n\) or \(\mathrm {R}^{n\times n}\). Additionally, we recall that
equipped with the usual norm \(\Vert {\varvec{\tau }}\Vert ^2_{\mathbf {div};\Omega } := \Vert {\varvec{\tau }}\Vert ^2_{0,\Omega } + \Vert \mathbf {div}({\varvec{\tau }})\Vert ^2_{0,\Omega }\), is a standard Hilbert space in the realm of mixed problems. In addition, \(\mathrm {H}^{1/2}(\Gamma )\) is the space of traces of functions of \(\mathrm {H}^1(\Omega )\) and \(\mathrm {H}^{-1/2}(\Gamma )\) denotes its dual. Also, by \(\left<\cdot ,\cdot \right>_{\Gamma }\) we will denote the corresponding product of duality between \(\mathrm {H}^{-1/2}(\Gamma )\) and \(\mathrm {H}^{1/2}(\Gamma )\) (and also between \({\mathbf {H}}^{-1/2}(\Gamma )\) and \({\mathbf {H}}^{1/2}(\Gamma )\)). Finally, throughout the rest of the paper we employ \({{\mathbf {0}}}\) to denote a generic null vector (or tensor), and use C and c, with or without subscripts, bars, tildes or hats, to denote generic constants independent of the discretization parameters, which may take different values at different places.
2 The Continuous Formulation
In this section we introduce the model problem and derive the corresponding weak formulation.
2.1 The Model Problem
In what follows we consider the model introduced in [28], which is given by a steady double-diffusive convection system in a fluid saturated porous medium. More precisely, we focus on solving the coupling of the incompressible Brinkman–Forchheimer and double-diffusion equations, which reduces to finding a velocity field \({\mathbf {u}}\), a pressure field p, a temperature field \(\phi _1\) and a concentration field \(\phi _2\), both defining a vector \(\underline{\varvec{\phi }}:= (\phi _1, \phi _2)\), such that
with parameters \(\nu := \mathtt {D}_a\,\widetilde{\mu }/\mu \) and \(\mathtt {F}:= \vartheta \,\mathtt {D}_a\,\mathtt {R}_1\), where \(\mathtt {D}_a\) stands for the Darcy number, \(\widetilde{\mu }\) the viscosity, \(\mu \) the effective viscosity, \(\mathtt {R}_1\) the thermal Rayleigh number, \(\mathtt {R}_2\) the solute Rayleigh number, and \(\vartheta \) is a real number that can be calculated experimentally. In addition, the external force \({\mathbf {f}}\) is defined by
with \({\mathbf {g}}\) representing the potential type gravitational acceleration, \(\phi _{1,\mathtt {r}}\) the reference temperature, \(\phi _{2,\mathtt {r}}\) the reference concentration of a solute, both of them living in \(\mathrm {L}^4(\Omega )\), and \(\varrho \) is another parameter experimentally valued that can be assumed to be \(\ge 1\) (see [28, Sect. 2] for details). In turn, the permeability, thermal diffusion and concentration diffusion tensors are denoted, respectively, by \({\mathbf {K}}, {\mathbf {Q}}_1\) and \({\mathbf {Q}}_2\) living in \({\mathbb {L}}^\infty (\Omega )\). Moreover, \({\mathbf {K}}\) and the inverses of \({\mathbf {Q}}_1\) and \({\mathbf {Q}}_2\), are uniformly positive definite tensors, which means that there exist positive constants \(C_{{\mathbf {K}}}\), \(C_{{\mathbf {Q}}_1}\), and \(C_{{\mathbf {Q}}_2}\), such that
Equations (2.1) are complemented with Dirichlet boundary conditions for the velocity, the temperature, and the concentration fields, that is
with given data \({\mathbf {u}}_\mathrm {D}\in {\mathbf {H}}^{1/2}(\Gamma )\), \(\phi _{1,\mathrm {D}}\in \mathrm {H}^{1/2}(\Gamma )\) and \(\phi _{2,\mathrm {D}}\in \mathrm {H}^{1/2}(\Gamma )\). Owing to the incompressibility of the fluid and the Dirichlet boundary condition for \({\mathbf {u}}\), the datum \({\mathbf {u}}_\mathrm {D}\) must satisfy the compatibility condition
In addition, due to the first equation of (2.1), and in order to guarantee uniqueness of the pressure, this unknown will be sought in the space
Next, in order to derive a fully-mixed formulation for (2.1)–(2.4), in which the Dirichlet boundary conditions become natural ones, we now proceed as in [11] (see similar approaches in [17, 18]), and introduce as further unknowns the pseudostress tensor \({\varvec{\sigma }}\), the pseudoheat vector \({\varvec{\rho }}_1\), and the pseudodiffusive vector \({\varvec{\rho }}_2\), which are defined by
In this way, applying the trace operator to \({\varvec{\sigma }}\) and utilizing the incompressibility condition \(\mathrm {div}({\mathbf {u}}) = 0\) in \(\Omega \), one arrives at
Hence, replacing back (2.7) in the first equation of (2.6), we find that our model problem (2.1)–(2.4) can be rewritten, equivalently, as follows: Find \(({\varvec{\sigma }}, {\mathbf {u}})\) and \(({\varvec{\rho }}_j, \phi _j)\), \(j\in \{1,2\}\), in suitable spaces to be indicated below such that
where the Dirichlet datum for \(\underline{\varvec{\phi }}\) is certainly given by \(\underline{\varvec{\phi }}_{\mathrm {D}} := (\phi _{1,\mathrm {D}}, \phi _{2,\mathrm {D}})\). At this point we stress that, as suggested by (2.7), p is eliminated from the present formulation and computed afterwards in terms of \({\varvec{\sigma }}\) by using that identity. This fact, justifies the last equation in (2.8), which aims to ensure that the resulting p does belong to \(\mathrm {L}^2_0(\Omega )\). Notice also that further variables of interest, such as the velocity gradient \(\nabla {\mathbf {u}}\), the heat vector \(\widetilde{{\varvec{\rho }}}_1 := {\mathbf {Q}}_1\,\nabla \phi _1\) and the diffusive vector \(\widetilde{{\varvec{\rho }}}_2 := {\mathbf {Q}}_2\,\nabla \phi _2\), can be computed, respectively, as follows
2.2 The Fully-Mixed Variational Formulation
In this section we derive our fully-mixed formulation for the coupled system given by (2.8). To that end, we multiply the first equation of (2.8) by a tensor \({\varvec{\tau }}\in {\mathbb {H}}(\mathbf {div};\Omega )\), integrate the resulting expression by parts, and use the identity \({\varvec{\sigma }}^\mathrm {d}:{\varvec{\tau }}= {\varvec{\sigma }}^\mathrm {d}:{\varvec{\tau }}^\mathrm {d}\) and the Dirichlet boundary condition \({\mathbf {u}}= {\mathbf {u}}_\mathrm {D}\) on \(\Gamma \), to get
In order to have more flexibility for choosing the finite element subspaces, but at the same time avoiding the incorporation of new terms in the resulting variational equation, we now proceed similarly as in [24] (see also [25]), and replace \({\mathbf {u}}\) in the second term of the left-hand side of (2.10) by the expression arising from the second equation in (2.8), that is
In this way, we arrive at the variational formulation: Find \({\varvec{\sigma }}\in {\mathbb {H}}(\mathbf {div};\Omega )\) and \({\mathbf {u}}\) (in a suitable space to be specified below), such that
for all \({\varvec{\tau }}\in {\mathbb {H}}(\mathbf {div};\Omega )\). Since \({\mathbf {K}}\in {\mathbb {L}}^\infty (\Omega )\) and \(\mathbf {div}({\varvec{\tau }})\in {\mathbf {L}}^2(\Omega )\), the term \({\mathbf {K}}|{\mathbf {u}}|{\mathbf {u}}\cdot \mathbf {div}({\varvec{\tau }})\) forces the velocity \({\mathbf {u}}\), and consequently the test function \({{\mathbf {v}}}\), to live in \({\mathbf {L}}^4(\Omega )\). In order to deal with this fact, we first observe, applying the Cauchy–Schwarz and Hölder inequalities, and then the continuous injection \({\mathbf {i}}_4\) of \({\mathbf {H}}^1(\Omega )\) into \({\mathbf {L}}^4(\Omega )\) (see, e.g., [32, Theorem 1.3.4]), that
for all \({{\mathbf {w}}},{\mathbf {u}}\in {\mathbf {H}}^1(\Omega )\) and \({\varvec{\tau }}\in {\mathbb {H}}(\mathbf {div};\Omega )\). However, we notice from (2.11) that the lack of a test function in the space where \({\mathbf {u}}\) lives (now in \({\mathbf {H}}^1(\Omega )\)), makes the well-posedness analysis of (2.11) non-viable. Then, aiming to circumvent this inconvenient, we propose to enrich our formulation with the following residual terms arising from the constitutive equation (first equation of (2.8)) and the Dirichlet boundary condition \({\mathbf {u}}= {\mathbf {u}}_\mathrm {D}\) on \(\Gamma \):
for all \({{\mathbf {v}}}\in {\mathbf {H}}^1(\Omega )\), where \(\kappa _1, \kappa _2\) are positive parameters to be specified later. We now recall (see, e.g., [4, 23, 26]) that there holds
where
Hence, decomposing \({\varvec{\tau }}\in {\mathbb {H}}(\mathbf {div};\Omega )\) as \({\varvec{\tau }}= {\varvec{\tau }}_0 + c\,{\mathbb {I}}\), with \({\varvec{\tau }}_0\in {\mathbb {H}}_0(\mathbf {div};\Omega )\) and \(c\in \mathrm {R}\), noticing that \({\varvec{\tau }}^\mathrm {d}= {\varvec{\tau }}^\mathrm {d}_0\) and \(\mathbf {div}({\varvec{\tau }}) = \mathbf {div}({\varvec{\tau }}_0)\), and using the last equation of (2.8) and the compatibility condition (2.5), we deduce that both \({\varvec{\sigma }}\) and \({\varvec{\tau }}\) can be considered hereafter in \({\mathbb {H}}_0(\mathbf {div};\Omega )\). Therefore, from (2.11) and (2.13), we arrive at the variational problem: Find \(({\varvec{\sigma }},{\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) such that
for all \(({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\), where given \({{\mathbf {w}}}\in {\mathbf {H}}^1(\Omega )\), A and \(B_{{\mathbf {w}}}\) are the bilinear forms defined, respectively, as
and
whereas, given \(\underline{\varvec{\varphi }}:= (\varphi _1,\varphi _2)\) in a suitable space defined next, \(F_\mathrm {D}\) and \(F_{\underline{\varvec{\varphi }}}\) are the bounded linear functionals defined by
On the other hand, for the double-diffusion equations in (2.8) we proceed as in [9] (see also [7, 15, 16]). In fact, multiplying the third and fourth equations of (2.8) by suitable test functions \({\varvec{\eta }}_j\) and \(\psi _j\), \(j\in \{1,2\}\), respectively, integrating by parts and using the Dirichlet boundary condition \(\underline{\varvec{\phi }}= \underline{\varvec{\phi }}_\mathrm {D}\) on \(\Gamma \), we get
for all \(({\varvec{\eta }}_j,\psi _j)\) in spaces to be derived below. In this regard, we begin by noting that for \({\mathbf {Q}}_j\in {\mathbb {L}}^\infty (\Omega )\), \(j\in \{1,2\}\), and \({\mathbf {u}}\in {\mathbf {H}}^1(\Omega )\), the first and third terms in the first equation of (2.18) are well defined if \({\varvec{\rho }}_j, \, {\varvec{\eta }}_j\in {\mathbf {L}}^2(\Omega )\), and if \(\phi _j\), and consequently the test function \(\psi _j\), are chosen to live in \(\mathrm {L}^4(\Omega )\), respectively. In this way, since the latter forces both \(\mathrm {div}({\varvec{\rho }}_j)\) and \(\mathrm {div}({\varvec{\eta }}_j)\) to live in \(\mathrm {L}^{4/3}(\Omega )\), we now introduce the Banach space
equipped with the norm
Notice that \({\mathbf {H}}(\mathrm {div};\Omega )\subset {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\). Moreover, as remarked in [9, Eq. (2.5)] (see also [15, Eq. (3.2)]), the right-hand side of (2.18) is well defined in the sense that \({\varvec{\eta }}_j\cdot {{\mathbf {n}}}\in \mathrm {H}^{-1/2}(\Gamma )\), \(j\in \{1,2\}\), for all \({\varvec{\eta }}_j\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\). Thus, the weak formulation for the double-diffusion equations in (2.8) reads: Find \(({\varvec{\rho }}_j,\phi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\), \(j\in \{1,2\}\), such that
where, given \({{\mathbf {w}}}\in {\mathbf {H}}^1(\Omega )\), \(a_j\), b, and \(c_j({{\mathbf {w}}};\cdot ,\cdot )\) are the forms defined, respectively, as
for all \({\varvec{\rho }}_j, {\varvec{\eta }}_j\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\) and \(\psi _j\in \mathrm {L}^4(\Omega )\). In turn, \(G_j\) is the bounded linear functional defined by
Then, the augmented fully-mixed formulation for the coupled problem (2.8) reduces to (2.14) and (2.19), that is: Find \(({\varvec{\sigma }}, {\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) and \(({\varvec{\rho }}_j, \phi _j) \in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\), \(j\in \{1, 2\}\), such that
for all \(({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) and for all \(({\varvec{\eta }}_j, \psi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\).
3 Analysis of the Coupled Problem
In this section we combine the Lax–Milgram, Banach–Nečas–Babuška, and Babuška–Brezzi theories, with the classical Schauder and Banach fixed-point theorems, to prove the well-posedness of (2.22) under suitable smallness assumptions on the data.
3.1 Preliminaries
We begin by recalling the Banach–Nečas–Babuška theorem, which is the Banach version of the generalized Lax–Milgram lemma in Hilbert spaces (see for instance [23, Theorem 1.1]). More precisely, we have the following result [20, Theorem 2.6].
Theorem 3.1
Let \(\mathrm {H}\) be a reflexive Banach space, and let \(a:\mathrm {H}\times \mathrm {H}\rightarrow \mathrm {R}\) be a bounded bilinear form. In addition, assume that
-
(i)
there exists \(\alpha > 0\) such that
$$\begin{aligned} \sup _{0\ne v\in \mathrm {H}}\, \frac{a(u,v)}{\Vert v\Vert _{\mathrm {H}}} \,\ge \, \alpha \,\Vert u\Vert _{\mathrm {H}}\quad \forall \, u\in \mathrm {H}, \end{aligned}$$(3.1) -
(ii)
there holds
$$\begin{aligned} \sup _{u\in \mathrm {H}}\,a(u,v) \,>\, 0 \quad \forall \,v\in \mathrm {H},\, v\ne 0. \end{aligned}$$(3.2)
Then, for each \(F\in \mathrm {H}'\) there exists a unique \(u\in \mathrm {H}\) such that \(\,a(u,v) = F(v) \quad \forall \, v\in \mathrm {H}\), and the following a priori estimate holds:
Let us now discuss the stability properties of the forms involved in (2.22). In fact, using (2.12) and performing simple computations, we deduce from (2.15), (2.16), and (2.20) that the forms \(A, B_{{{\mathbf {w}}}}, a_j, b\) and \(c_j({{\mathbf {w}}};\cdot ,\cdot )\), \(j\in \{1,2\}\), are bounded as indicated in what follows
and
where \(C_A\) is a positive constant depending on \(\nu , \Vert {\mathbf {K}}\Vert _{\infty }, \kappa _1\), and \(\kappa _2\). In addition, employing the Cauchy–Schwarz and Young inequalities, and recalling the definition of \({\mathbf {f}}\) (cf. (2.2)), it is readily seen that \(F_\mathrm {D}, F_{\underline{\varvec{\varphi }}}\) and \(G_j\) (cf. (2.21)) are bounded, which means that there exist constants \(C_{\mathrm {D}}\), \(C_F\), \(C_G >0\), such that
and
where \(C_\mathrm {D}:= \max \big \{1,\kappa _2\,\Vert \gamma _0\Vert \big \}\), \(C_F := \Vert {\mathbf {K}}\Vert _\infty \), \(\underline{\varvec{\phi }}_\mathtt {r}:= (\phi _{1,\mathtt {r}}, \phi _{2,\mathtt {r}})\in {\mathbf {L}}^4(\Omega )\), and \(C_{G_j}\) is a positive constant depending on \(\Vert {\mathbf {i}}_{4}\Vert \) (cf. [7, Lemma 3.5]). Next, we let \({\mathbf {V}}\) be the kernel of the operator induced by the bilinear form b, that is
and observe, thanks to the definition of \(a_j\) (cf. (2.20)) and the fact that the inverses of \({\mathbf {Q}}_j\) are uniformly positive definite tensors (cf. (2.3)), that \(a_j\) is elliptic on \({\mathbf {V}}\), that is
with \(\alpha _j = C_{{\mathbf {Q}}_j}\). In turn, according to [9, Lemma 2.1] with \(p = 4/3\), we know that there exists a constant \(\beta > 0\) such that b verify the following inf-sup condition
We end this section by recalling, for later use, that there exist positive constants \(c_1(\Omega )\) and \(c_2(\Omega )\) such that (see [23, Lemma 2.3] and [29, Theorem 5.11.2], respectively, for details)
and
3.2 A Fixed-Point Approach
We now rewrite (2.22) as an equivalent fixed-point equation. To this end, we first let \({\mathbf {S}}: {\mathbf {H}}^1(\Omega )\times {\mathbf {L}}^4(\Omega )\rightarrow {\mathbf {H}}^1(\Omega )\) be the operator defined as
where \(({\varvec{\sigma }},{\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) is the unique solution (to be confirmed below) of the problem:
for all \(({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\). In turn, for each \(j\in \{1,2\}\) we let \(\widetilde{{\mathbf {S}}}_j: {\mathbf {H}}^1(\Omega )\rightarrow \mathrm {L}^4(\Omega )\) be the operator given by
where \(({\varvec{\rho }}_j,\phi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\) is the unique solution (to be confirmed below) of the problem:
so that we can introduce \(\widetilde{{\mathbf {S}}}({{\mathbf {w}}}) := (\widetilde{{\mathbf {S}}}_1({{\mathbf {w}}}), \widetilde{{\mathbf {S}}}_2({{\mathbf {w}}}))\in {\mathbf {L}}^4(\Omega )\) for all \({{\mathbf {w}}}\in {\mathbf {H}}^1(\Omega )\). Consequently, we can define the operator \({\mathbf {T}}: {\mathbf {H}}^1(\Omega )\rightarrow {\mathbf {H}}^1(\Omega )\) as
and realize that solving (2.22) is equivalent to finding \({\mathbf {u}}\in {\mathbf {H}}^1(\Omega )\) such that
3.3 Well-Definedness of \({\mathbf {T}}\)
We begin by establishing a result that provides sufficient conditions under which the operator \({\mathbf {S}}\) (cf. (3.15)) is well-defined, or equivalently, the problem (3.16) is well-posed.
Lemma 3.2
Assume that for \(\delta \in (0,2\nu )\) we choose \(\kappa _1\in (0,2\delta )\) and \(\kappa _2>0\). Then, there exists \(r_1>0\) such that for each \(r\in (0,r_1)\), and for each \(({{\mathbf {w}}}, \underline{\varvec{\varphi }})\in {\mathbf {H}}^1(\Omega )\times {\mathbf {L}}^4(\Omega )\) satisfying \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }\le r\), the problem (3.16) has a unique solution \(({\varvec{\sigma }},{\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\). Moreover, there exists a constant \(c_{\mathbf {S}}> 0\), independent of \(({{\mathbf {w}}},\underline{\varvec{\varphi }})\), such that there holds
Proof
We proceed as in [11, Lemma 3.2]. In fact, given \(({{\mathbf {w}}},\underline{\varvec{\varphi }})\in {\mathbf {H}}^1(\Omega )\times {\mathbf {L}}^4(\Omega )\), we observe from (2.15) and (2.16) that \(A + B_{{\mathbf {w}}}\) is clearly a bilinear form. Then, thanks to (3.3) and (3.4), we find that there exists a positive constant, which we denote by \(\Vert A + B_{{\mathbf {w}}}\Vert \), only depending on \(\nu , \Vert {\mathbf {K}}\Vert _\infty , \Vert {\mathbf {i}}_4\Vert , \kappa _1, \kappa _2, \mathtt {F}\), and \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }\), such that
for all \(({\varvec{\sigma }},{\mathbf {u}}), ({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\). In turn, from the definition of A (cf. (2.15)), we have
and hence, using (2.3), and Cauchy–Schwarz and Young’s inequalities, we obtain that for any \(\delta >0\) and for all \(({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\), there holds
In this way, applying the inequalities (3.13) and (3.14), we can define the constants
which are positive thanks to the hypotheses on \(\delta , \kappa _1\), and \(\kappa _2\). In this way, it follows that
with \(\alpha _A := \min \Big \{\alpha _1(\Omega ), \alpha _2(\Omega )\Big \}\), which shows that A is elliptic on \({\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\). Therefore, combining now (3.4) and (3.23), and using the injection \({\mathbf {i}}_4\), we deduce that for all \(({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) there holds
Consequently, requiring \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega } \le r_1\), with
we arrive at
Summing up, and owing to the hypotheses on \(\kappa _1\) and \(\kappa _2\), we have proved that for any \(({{\mathbf {w}}},\underline{\varvec{\varphi }})\in {\mathbf {H}}^1(\Omega )\times {\mathbf {L}}^4(\Omega )\) such that \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega } \le r_1\), the bilinear form \(A + B_{{\mathbf {w}}}\) and the functional \(F_\mathrm {D}+ F_{\underline{\varvec{\varphi }}}\) satisfy the hypotheses of the Lax–Milgram theorem (see, e.g., [23, Theorem 1.1]), which guarantees the well-posedness of (3.16). Finally, using (3.25) with \(({\varvec{\tau }},{{\mathbf {v}}}) = ({\varvec{\sigma }},{\mathbf {u}})\), (3.16), and the bounds of \(F_\mathrm {D}\) and \(F_{\underline{\varvec{\varphi }}}\) (cf. (3.8) and (3.9)), we readily obtain that
which implies (3.21) with \(c_{{\mathbf {S}}} := 2\,\max \big \{ C_\mathrm {D}, C_F \big \}/\alpha _A\), thus completing the proof. \(\square \)
Now, we establish the well-posedness of problem (3.18), or equivalently, that the operator \(\widetilde{{\mathbf {S}}}\) (cf. (3.17)) is well-defined. To that end, let us consider the space \({\mathbf {H}}:= {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\) and the bilinear form \({\mathcal {A}}_j : {\mathbf {H}}\times {\mathbf {H}}\rightarrow \mathrm {R}\), \(j\in \{1,2\}\), defined by
for all \(({\varvec{\rho }}_j,\phi _j),({\varvec{\eta }}_j,\psi _j)\in {\mathbf {H}}\). Then, owing to (3.5), (3.6), (3.11), (3.12), and a direct application of [20, Proposition 2.36], we deduce, equivalently, that \({\mathcal {A}}_j\) satisfies the following inf-sup condition:
where \(\gamma _j > 0\) is the constant defined by
In this way, bearing in mind (3.27), we are able to establish the following result.
Lemma 3.3
There exists \(r_2>0\) such that for each \(r\in (0,r_2)\), for each \({{\mathbf {w}}}\in {\mathbf {H}}^1(\Omega )\) satisfying \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega } \le r\), and for each \(j\in \{1,2\}\), the problem (3.18) has a unique solution \(({\varvec{\rho }}_j,\phi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\), \(j\in \{1,2\}\). Moreover, there exists a constant \(c_{\widetilde{{\mathbf {S}}}} > 0\), independent of \({{\mathbf {w}}}\), such that there holds
Proof
We proceed as in [7, Theorem 3.6]. In fact, given \({{\mathbf {w}}}\in {\mathbf {H}}^1(\Omega )\) and \(j\in \{1,2\}\), we begin by defining the bilinear form
where \({\mathcal {A}}_j\) and \(c_j({{\mathbf {w}}};\cdot ,\cdot )\) are the forms defined in (3.26) and (2.20), respectively. Then, the problem (3.18) can be rewritten, equivalently, as: Find \(({\varvec{\rho }}_j,\phi _j)\in {\mathbf {H}}:= {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\) such that
Therefore, in order to conclude the well-definedness of \(\widetilde{{\mathbf {S}}}\), in the sequel we use the Banach–Nečas–Babuška theorem (cf. Theoremm 3.1) to prove that problem (3.31) is well-posed. Indeed, given \(({\varvec{\chi }}_j,\varphi _j), (\widetilde{{\varvec{\eta }}}_j,\widetilde{\psi }_j)\in {\mathbf {H}}\) with \((\widetilde{{\varvec{\eta }}}_j,\widetilde{\psi }_j)\ne {{\mathbf {0}}}\), we first deduce from (3.30) and (3.7) that
which, together with (3.27) and the fact that \((\widetilde{{\varvec{\eta }}}_j,\widetilde{\psi }_j)\in {\mathbf {H}}\) is arbitrary, implies
Consequently, requiring now \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega } \le r_2\), with
we find that
On the other hand, for a given \(({\varvec{\chi }}_j,\varphi _j)\in {\mathbf {H}}\), we observe that
which, employing again (3.7), yields
Therefore, using the symmetry of \({\mathcal {A}}_j\) and the inf-sup condition (3.27), and considering \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }\le r_2\) (cf. (3.32)), we obtain
In this way, it is clear from (3.33) and (3.34) that \({\mathcal {A}}_{j,{{\mathbf {w}}}}\) satisfies the hypotheses of the Banach–Nečas–Babuška theorem (cf. Theorem 3.1), which guarantees the well-posedness of (3.18). Finally, using (3.33) with \(({\varvec{\chi }}_j,\varphi _j) = ({\varvec{\rho }}_j, \phi _j)\), (3.31), and the continuity bound of \(G_j\) (cf. (3.10)), we get
which gives (3.29) with \(\displaystyle c_{\widetilde{{\mathbf {S}}}} := \max \big \{ c_{\widetilde{{\mathbf {S}}}_1}, c_{\widetilde{{\mathbf {S}}}_2} \big \}\) and \(c_{\widetilde{{\mathbf {S}}}_j} := 2\,C_{G_j}/\gamma _j\), \(j\in \{1,2\}\), thus ending proof. \(\square \)
Now, concerning the practical choice of the stabilization parameters \(\kappa _1\) and \(\kappa _2\), and particularly for sake of the computational implementation of the Galerkin method to be introduced and analyzed later on, we first select the midpoints of the corresponding feasible intervals for \(\delta \) and \(\kappa _1\), namely \(\delta = \nu \) and \(\kappa _1 = \delta \), respectively. Then, in order to define \(\kappa _2\), we aim to maximize the constant \(\alpha _2(\Omega )\) in (3.22), which is attained by taking \(\kappa _2 = \kappa _1\left( 1-\frac{\delta }{2\nu }\right) \). In this way, we propose to consider:
3.4 Solvability Analysis of the Fixed-Point Equation
Having proved the well-posedness of the uncoupled problems (3.16) and (3.18), which ensures that the operators \({\mathbf {S}}, \widetilde{{\mathbf {S}}}\) and \({\mathbf {T}}\) are well defined, we now aim to establish the existence of a unique fixed point of the operator \({\mathbf {T}}\). For this purpose, in what follows we verify the hypothesis of the Schauder and Banach fixed-point theorems. We begin the analysis with the following straightforward consequence of Lemmas 3.2 and 3.3.
Lemma 3.4
Given \(r\in (0,r_0)\), with \(r_0 := \min \{r_1,r_2\}\), \(r_1\) as in (3.24) and \(r_2\) as in (3.32), we let \({\mathbf {W}}_r\) be the closed and convex subset of \({\mathbf {H}}^1(\Omega )\) defined by
In addition, we take the stabilization parameters \(\kappa _1\) and \(\kappa _2\) as in Lemma 3.2 (particularly, as suggested in (3.36)), and assume that the data satisfy
Then \({\mathbf {T}}({\mathbf {W}}_r) \,\subseteq \, {\mathbf {W}}_r\).
Proof
Given \({{\mathbf {w}}}\in {\mathbf {W}}_r\), we let \(\,\underline{\varvec{\phi }}\,:=\,\widetilde{{\mathbf {S}}}({{\mathbf {w}}})\) and observe that certainly \(({{\mathbf {w}}},\underline{\varvec{\phi }})\) satisfies the hypotheses of Lemma 3.2. Hence, employing the corresponding estimate (3.21) in combination with (3.29), we get
with \(c_{{\mathbf {T}}} := c_{{\mathbf {S}}} \max \big \{ 1, c_{\widetilde{{\mathbf {S}}}} \big \}\). In this way, (3.39) and assumption (3.38) imply that \(\,{\mathbf {T}}({{\mathbf {w}}}) \in {\mathbf {W}}_r\), which proves that \({\mathbf {T}}({\mathbf {W}}_r) \subseteq {\mathbf {W}}_r\). \(\square \)
Next, we aim to prove the continuity and compactness properties of \({\mathbf {T}}\), which basically will be direct consequences of the following two auxiliary lemmas for the operators \({\mathbf {S}}\) and \(\widetilde{{\mathbf {S}}}\), respectively.
Lemma 3.5
Let \(r\in (0,r_1)\) with \(r_1\) given as in (3.24). Then there exists a positive constant \(C_{\mathbf {S}}\) independent of r, such that
for all \(({{\mathbf {w}}},\underline{\varvec{\varphi }}), \,(\widetilde{{{\mathbf {w}}}},\widetilde{\underline{\varvec{\varphi }}})\in {\mathbf {H}}^1(\Omega )\times {\mathbf {L}}^4(\Omega )\) such that \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }, \Vert \widetilde{{{\mathbf {w}}}}\Vert _{1,\Omega } \le r\).
Proof
Given \(({{\mathbf {w}}},\underline{\varvec{\varphi }}), (\widetilde{{{\mathbf {w}}}},\widetilde{\underline{\varvec{\varphi }}})\in {\mathbf {H}}^1(\Omega )\times {\mathbf {L}}^4(\Omega )\), such that \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }, \Vert \widetilde{{{\mathbf {w}}}}\Vert _{1,\Omega } \le r\), we let \(({\varvec{\sigma }},{\mathbf {u}}), (\widetilde{{\varvec{\sigma }}},\widetilde{{\mathbf {u}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) be the corresponding solutions of (3.16), so that \({\mathbf {u}}:= {\mathbf {S}}({{\mathbf {w}}},\underline{\varvec{\varphi }})\) and \(\widetilde{{\mathbf {u}}} := {\mathbf {S}}(\widetilde{{{\mathbf {w}}}},\widetilde{\underline{\varvec{\varphi }}})\). Then, using the bilinearity of A and \(B_{{\mathbf {w}}}\) for any \({{\mathbf {w}}}\), it follows easily from (3.16) that
for all \(({\varvec{\tau }},{{\mathbf {v}}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\). Hence, taking \(\,({\varvec{\tau }},{{\mathbf {v}}}) = ({\varvec{\sigma }},{\mathbf {u}}) - (\widetilde{{\varvec{\sigma }}},\widetilde{{\mathbf {u}}})\) in the foregoing identity, and then employing the ellipticity of \(A+B_{{\mathbf {w}}}\) (cf. (3.25)), the fact that \(|\,|\widetilde{{{\mathbf {w}}}}| - |{{\mathbf {w}}}|\,| \le |\widetilde{{{\mathbf {w}}}} - {{\mathbf {w}}}|\), the boundedness property of \(B_{{\mathbf {w}}}\) (cf. (3.4)), and the definition and continuity of \(F_{\underline{\varvec{\varphi }}}\) (cf. (2.17), (3.9)) in combination with Cauchy–Schwarz and Hölder’s inequalities, we readily get
which, together with the definition of \(r_1\) (cf. (3.24)) and the fact that \({\mathbf {u}}= {\mathbf {S}}({{\mathbf {w}}},\underline{\varvec{\varphi }})\), implies (3.40) with constant \(C_{{\mathbf {S}}} := 2\,C_F/\alpha _{A}\), thus ending the proof. \(\square \)
Lemma 3.6
Let \(r\in (0,r_2)\) with \(r_2\) given as in (3.32). Then there holds
for all \({{\mathbf {w}}}, \,\widetilde{{{\mathbf {w}}}} \in {\mathbf {H}}^1(\Omega )\) such that \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }, \, \Vert \widetilde{{{\mathbf {w}}}}\Vert _{1,\Omega } \le r\).
Proof
We prooced as in [7, Theorem 3.7]. In fact, given \({{\mathbf {w}}}, \widetilde{{{\mathbf {w}}}} \in {\mathbf {H}}^1(\Omega )\) such that \(\Vert {{\mathbf {w}}}\Vert _{1,\Omega }, \, \Vert \widetilde{{{\mathbf {w}}}}\Vert _{1,\Omega } \le r\), for each \(j\in \{1,2\}\) we let \(({\varvec{\rho }}_j, \phi _j), \, (\widetilde{{\varvec{\rho }}}_j, \widetilde{\phi }_j) \in {\mathbf {H}}:= {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\) be the solution of (3.18) (equivalently of (3.31)), so that \((\phi _1, \phi _2) = (\widetilde{{\mathbf {S}}}_1({{\mathbf {w}}}),\widetilde{{\mathbf {S}}}_2({{\mathbf {w}}})) = \widetilde{{\mathbf {S}}}({{\mathbf {w}}})\) and \((\widetilde{\phi }_1, \widetilde{\phi }_2) = (\widetilde{{\mathbf {S}}}_1(\widetilde{{{\mathbf {w}}}}), \widetilde{{\mathbf {S}}}_2(\widetilde{{{\mathbf {w}}}})) = \widetilde{{\mathbf {S}}}(\widetilde{{{\mathbf {w}}}})\). Then, using the linearity of the form \({\mathcal {A}}_{j,{{\mathbf {w}}}}\) (cf. (3.30)), we deduce after simple computations that
Then, employing (3.33) with \(({\varvec{\chi }}_j, \varphi _j) = ({\varvec{\rho }}_j - \widetilde{{\varvec{\rho }}}_j, \phi _j - \widetilde{\phi }_j)\in {\mathbf {H}}\) and the continuity bound of \(c_j({{\mathbf {w}}};\cdot ,\cdot )\) (cf. (3.7)), we obtain
which, together with the definition of \(r_2\) (cf. (3.32)) and the fact that \(\phi _j = \widetilde{{\mathbf {S}}}_j({{\mathbf {w}}})\), implies (3.41) and conclude the proof. \(\square \)
As a consequence of Lemmas 3.5 and 3.6 we establish the following result providing an estimate needed to derive next the required continuity and compactness properties of the operator \({\mathbf {T}}\).
Lemma 3.7
Let \(r\in (0,r_0)\), with \(r_0 := \min \{r_1,r_2\}\), \(r_1\) as in (3.24) and \(r_2\) as in (3.32). Then, for all \({{\mathbf {w}}}, \widetilde{{{\mathbf {w}}}} \in {\mathbf {W}}_r\) (cf. (3.37)), there holds
Proof
Let \({{\mathbf {w}}}, \widetilde{{{\mathbf {w}}}}\in {\mathbf {W}}_r\), such that \({\mathbf {u}}={\mathbf {T}}({{\mathbf {w}}})\) and \(\widetilde{{\mathbf {u}}} = {\mathbf {T}}(\widetilde{{{\mathbf {w}}}})\). Then, from the definition of \({\mathbf {T}}\) (cf. (3.19)), and Lemmas 3.5 and 3.6 (cf. (3.40) and (3.41)), we deduce that
Hence, using (3.29) and the fact that \(C_{{\mathbf {S}}}\,c_{\widetilde{{\mathbf {S}}}}\) is bounded by \(c_{\mathbf {T}}\), and then bounding \(\Vert {\mathbf {T}}({{\mathbf {w}}})\Vert _{1,\Omega }\) by (3.39) instead of by r, we conclude from the foregoing inequality that
Finally, (3.42) follows from (3.43) by noting that both \(1/r_1\) and \(1/r_2\) are bounded by \(1/r_0\). \(\square \)
Owing to the above analysis, we establish now the announced properties of the operator \({\mathbf {T}}\).
Lemma 3.8
Let \(r\in (0,r_0)\), with \(r_0 := \min \{r_1,r_2\}\), \(r_1\) as in (3.24) and \(r_2\) as in (3.32). Assume that the stabilization parameters \(\kappa _1\) and \(\kappa _2\) are taken as in Lemma 3.2, and that the data satisfy (3.38). Then \({\mathbf {T}}: {\mathbf {W}}_r \rightarrow {\mathbf {W}}_r\) is continuous and \(\overline{{\mathbf {T}}({\mathbf {W}}_r)}\) is compact.
Proof
The required result follows straightforwardly from estimate (3.42), the compactness of the injection \({\mathbf {i}}_4 :{\mathbf {H}}^1(\Omega )\rightarrow {\mathbf {L}}^4(\Omega )\) (see, e.g., [32, Theorem 1.3.5]), and the well-known fact that every bounded sequence in a Hilbert space has a weakly convergent subsequence. We omit further details and refer to [8, Lemma 3.8]. \(\square \)
Finally, the main result of this section is stated as follows.
Theorem 3.9
Assume the same hypothesis of Lemma 3.8. Then the operator \({\mathbf {T}}\) has a fixed point \({\mathbf {u}}\in {\mathbf {W}}_r\) (cf. (3.37)). Equivalently, the coupled problem (2.22) has a solution \(({\varvec{\sigma }}, {\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) and \(({\varvec{\rho }}_j, \phi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\), \(j\in \{1,2\}\), with \({\mathbf {u}}\in {\mathbf {W}}_r\). Moreover, there holds
and
In addition, if the data satisfy
then the aforementioned fixed point (equivalently, the solution of (2.22)) is unique.
Proof
The equivalence between (2.22) and the fixed point equation (3.20), together with Lemmas 3.4 and 3.8, confirm the existence of solution of (2.22) as a direct application of the Schauder fixed-point theorem [14, Theorem 9.12-1(b)]. In addition, it is clear that the estimate (3.45) follows from (3.29), whereas combining (3.21) with (3.29) we obtain (3.44) (cf. (3.39)). On the other hand, using the estimate (3.42) and the continuous injection \({\mathbf {i}}_4\) of \({\mathbf {H}}^1(\Omega )\) into \({\mathbf {L}}^4(\Omega )\), we easily obtain
which, thanks to (3.46) and the Banach fixed-point theorem, yields the uniqueness. \(\square \)
We end this section by remarking that we are able to avoid the augmentation procedure developed for the Brinkman–Forchheimer equations by proceeding as in [15], that is, introducing the gradient of the velocity as a new unknown of the system (besides the pseudostress tensor and the velocity of the fluid), which yields a new three-field Banach mixed formulation. We are addressing this issue in an ongoing work [10].
4 The Galerkin Scheme
In this section, we introduce and analyze the corresponding Galerkin scheme for the fully-mixed formulation (2.22). The solvability of this scheme is addressed following analogous tools to those employed throughout Sect. 3.
4.1 Preliminaries
We begin by considering arbitrary finite dimensional subspaces
whose specific choices are postponed to Sect. 4.3 below. Hereafter, \(h := \max \big \{ h_T :\, T\in {\mathcal {T}}_h \}\) stands for the size of a regular triangulation \({\mathcal {T}}_h\) of \(\overline{\Omega }\) made up of triangles T (when \(n = 2\)) or tetrahedra T (when \(n = 3\)) of diameter \(h_T\). In what follows, we set \(\underline{\varvec{\phi }}_h := (\phi _{1,h}, \, \phi _{2,h}),\,\underline{\varvec{\varphi }}_h := (\varphi _{1,h}, \varphi _{2,h})\) \(\in {\mathbf {H}}^{\phi }_h := \mathrm {H}^{\phi }_h\times \mathrm {H}^{\phi }_h\). Then the Galerkin scheme associated with (2.22) reads: Find \(({\varvec{\sigma }}_h, {\mathbf {u}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_{h}\times {\mathbf {H}}^{{\mathbf {u}}}_h\) and \(({\varvec{\rho }}_{j,h}, \phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\), \(j\in \{1,2\}\), such that
for all \(({\varvec{\tau }}_h,{{\mathbf {v}}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\) and \(({\varvec{\eta }}_{j,h}, \psi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h \times \mathrm {H}^{\phi }_h\). We now develop the discrete analogue of the fixed-point approach utilized in Sect. 3.2. To this end, we first consider the operator \({\mathbf {S}}_h : {\mathbf {H}}^{{\mathbf {u}}}_h \times {\mathbf {H}}^{\phi }_h \rightarrow {\mathbf {H}}^{{\mathbf {u}}}_h\) defined by
where \(({\varvec{\sigma }}_h,{\mathbf {u}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\) is the unique solution (to be confirmed below) of the problem:
for all \(({\varvec{\tau }}_h,{{\mathbf {v}}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\). In turn, for each \(j\in \{1,2\}\) we let \(\widetilde{{\mathbf {S}}}_{j,h}: {\mathbf {H}}^{{\mathbf {u}}}_h \rightarrow \mathrm {H}^{\phi }_h\) be the operator given by
where \(({\varvec{\rho }}_{j,h},\phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\) is the unique solution (to be confirmed below) of the problem:
and then we set \(\widetilde{{\mathbf {S}}}_h({{\mathbf {w}}}_h) := (\widetilde{{\mathbf {S}}}_{1,h}({{\mathbf {w}}}_h),\widetilde{{\mathbf {S}}}_{2,h}({{\mathbf {w}}}_h))\in {\mathbf {H}}^{\phi }_h\) for all \({{\mathbf {w}}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\). Hence, introducing the operator \({\mathbf {T}}_h : {\mathbf {H}}^{{\mathbf {u}}}_h\rightarrow {\mathbf {H}}^{{\mathbf {u}}}_h\) as
we realize that solving (4.2) is equivalent to seeking a fixed point of \({\mathbf {T}}_h\), that is: Find \({\mathbf {u}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\) such that
4.2 Solvability Analysis
We begin by remarking that the same tools employed in the proof of Lemma 3.2 can be used now to prove the unique solvability of (4.4). In fact, it is straightforward to see that for each \({{\mathbf {w}}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\), the bilinear form \(A + B_{{{\mathbf {w}}}_h}\) is bounded with a constant depending on \(\nu \), \(\mathtt {F}\), \(\Vert {\mathbf {K}}\Vert _{\infty }\), \(\Vert {\mathbf {i}}_4\Vert \), \(\kappa _1\), \(\kappa _2\), and \(\Vert {{\mathbf {w}}}_h\Vert _{1,\Omega }\). In addition, under the same assumptions from Lemma 3.2 on the stabilization parameters, we find that for each \({{\mathbf {w}}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\), \(A + B_{{{\mathbf {w}}}_h}\) is elliptic on \({\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\) with the same constant obtained in (3.25). In turn, it is clear that for each \(\underline{\varvec{\varphi }}_h \in {\mathbf {H}}^{\phi }_h\), the functional \(F_{\underline{\varvec{\varphi }}_h}\) is linear and bounded as in (3.9). The foregoing discussion and the Lax–Milgram theorem allow us to conclude the following result.
Lemma 4.1
Let \(\widetilde{r}\in (0,r_1)\), with \(r_1\) given as in (3.24), and assume that \(\kappa _1\) and \(\kappa _2\) are taken as in Lemma 3.2. Then, for each \(({{\mathbf {w}}}_h, \underline{\varvec{\varphi }}_h)\in {\mathbf {H}}^{{\mathbf {u}}}_h\times {\mathbf {H}}^{\phi }_h\) satisfying \(\Vert {{\mathbf {w}}}_h\Vert _{1,\Omega } \le \widetilde{r}\), the problem (4.4) has a unique solution \(({\varvec{\sigma }}_h,{\mathbf {u}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\). Moreover, with the same constant \(c_{{\mathbf {S}}} > 0\) from (3.21), which is independent of \(({{\mathbf {w}}}_h,\underline{\varvec{\varphi }}_h)\), there holds
We emphasize here that there is no restriction on \({\mathbb {H}}^{{\varvec{\sigma }}}_h\) and \({\mathbf {H}}^{{\mathbf {u}}}_h\), and hence they can be chosen as any finite element subspaces of \({\mathbb {H}}_0(\mathbf {div};\Omega )\) and \({\mathbf {H}}^1(\Omega )\), respectively.
On the other hand, in order to analyze problem (4.6), we need to incorporate further hypotheses on the discrete spaces \({\mathbf {H}}^{{\varvec{\rho }}}_h\) and \(\mathrm {H}^{\phi }_h\). For this purpose, we now let \({\mathbf {V}}_{h}\) be the discrete kernel of b, i.e.,
Then, we assume that the following discrete inf-sup conditions hold:
(H.1) for each \(j\in \{1,2\}\) there exists a constant \(\widetilde{\alpha }_j > 0\), independent of h, such that
(H.2) there exists a constant \(\widetilde{\beta } > 0\), independent of h, such that
Specific examples of spaces verifying (H.1) and (H.2) are described later on in Sect. 4.3. Notice that (4.11) is more general, and hence less restrictive, than assuming that the bilinear forms \(a_j\) are elliptic in \({\mathbf {V}}_{h}\). In other words, the latter is not necessary but only a sufficient condition for (4.11), which is precisely what we apply below in Sect. 4.3 for a particular choice of subspaces. In turn, unless \({\mathbf {V}}_{h}\) is contained in \({\mathbf {V}}\), which occurs in many cases but not always, the \({\mathbf {V}}_{h}\)-ellipticity of \(a_j\) does not follow from its eventual \({\mathbf {V}}\)-ellipticity.
Next, we consider the bilinear form \({\mathcal {A}}_j\) (cf. (3.26)) restricted to the discrete space \({\mathbf {H}}_h := {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\). Thus, employing (3.5), (4.11), and (4.12), and applying again [20, Proposition 2.36], we are able to show that \({\mathcal {A}}_j\) verifies the following discrete inf-sup condition
where \(\widetilde{\gamma }_j\) (cf. (3.28)), a positive constant independent of h, is defined by
We are now in a position to establish the discrete version of Lemma 3.3.
Lemma 4.2
There exists \(\widetilde{r}_2 > 0\) such that for each \(\widetilde{r}\in (0,\widetilde{r}_2)\), and for each \({{\mathbf {w}}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\) satisfying \(\Vert {{\mathbf {w}}}_h\Vert _{1,\Omega } \le \widetilde{r}\), the problem (4.6) has a unique solution \(({\varvec{\rho }}_{j,h},\phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\), for each \(j\in \{1,2\}\). Moreover, there exists a constant \(c_{\widetilde{{\mathbf {S}}}_h} > 0\), independent of \({{\mathbf {w}}}_h\), such that there holds
Proof
Given \({{\mathbf {w}}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\), we proceed analogously to the proof of Lemma 3.3 and utilize the continuity bound of \(c_j\) (cf. (3.7)), and the discrete inf-sup condition of \({\mathcal {A}}_j\) (4.13), to deduce that for each \({{\mathbf {w}}}_h\in {\mathbf {H}}^{{\mathbf {u}}}_h\) satisfying \(\Vert {{\mathbf {w}}}_h\Vert _{1,\Omega }\le \widetilde{r}_2\), with
\({\mathcal {A}}_{j,{{\mathbf {w}}}_h}\) (cf. (3.30)) satisfies the discrete inf-sup condition
Therefore, owing to the fact that for finite dimensional linear problems, surjectivity and injectivity are equivalent, we conclude from (4.17) and Theorem 3.1 that for each \(j\in \{1,2\}\) there exists a unique \(({\varvec{\rho }}_{j,h}, \phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\) satisfying
which means that (4.6) is well-posed. In addition, proceeding similarly to (3.29) we obtain (4.15), with \(\displaystyle c_{\widetilde{{\mathbf {S}}}_h} := \max \big \{c_{\widetilde{{\mathbf {S}}}_{1,h}}, c_{\widetilde{{\mathbf {S}}}_{2,h}} \big \}\) and \(c_{\widetilde{{\mathbf {S}}}_{j,h}} := 2\,C_{G_j} / \widetilde{\gamma }_j\), \(j\in \{1,2\}\), which ends the proof. \(\square \)
We now proceed to analyze the fixed-point equation (4.8). We begin with the discrete version of Lemma 3.4, whose proof, being a simple translation of the arguments proving that lemma, is omitted.
Lemma 4.3
Let \(\widetilde{r}\in (0,\widetilde{r}_0)\), with \(\widetilde{r}_0 := \min \{r_1,\widetilde{r}_2\}\), \(r_1\) as in (3.24) and \(\widetilde{r}_2\) as in (4.16), and let \({\mathbf {W}}_{\widetilde{r}}\) be the bounded subset of \({\mathbf {H}}^{{\mathbf {u}}}_h\) defined by
Assume that \(\kappa _1\) and \(\kappa _2\) are taken as in Lemma 3.2, and that the data satisfy
where \(c_{{\mathbf {T}}_h} := c_{{\mathbf {S}}}\,\max \big \{ 1, c_{\widetilde{{\mathbf {S}}}_h} \big \}\). Then \({\mathbf {T}}_h({\mathbf {W}}_{\widetilde{r}}) \subseteq {\mathbf {W}}_{\widetilde{r}}\).
Next, we address the discrete counterparts of the auxiliary Lemmas 3.5 and 3.6, whose proofs, being almost verbatim of the continuous ones, are omitted. We just remark that Lemma 4.5 below is derived using the discrete inf-sup condition (4.13). Thus, we simply state the corresponding results as follows.
Lemma 4.4
Let \(\widetilde{r}\in (0,r_1)\) with \(r_1\) given as in (3.24). Then, with the same constant \(C_{{\mathbf {S}}}>0\) from (3.40), which is independent of \(\widetilde{r}\), there holds
for all \(({{\mathbf {w}}}_h,\underline{\varvec{\varphi }}_h), \, (\widetilde{{{\mathbf {w}}}}_h,\widetilde{\underline{\varvec{\varphi }}}_h) \in {\mathbf {H}}^{{\mathbf {u}}}_h \times {\mathbf {H}}^{\phi }_h\) such that \(\Vert {{\mathbf {w}}}_h\Vert _{1,\Omega }, \, \Vert \widetilde{{{\mathbf {w}}}}_h\Vert _{1,\Omega } \le \widetilde{r}\).
Lemma 4.5
Let \(\widetilde{r}\in (0,\widetilde{r}_2)\) with \(\widetilde{r}_2\) given as in (4.16). Then there holds
for all \({{\mathbf {w}}}_h, \,\widetilde{{{\mathbf {w}}}}_h \in {\mathbf {H}}^{{\mathbf {u}}}_h\) such that \(\Vert {{\mathbf {w}}}_h\Vert _{1,\Omega }, \, \Vert \widetilde{{{\mathbf {w}}}}_h\Vert _{1,\Omega } \le \widetilde{r}\).
As a straightforward consequence of Lemmas 4.4 and 4.5, and the continuous injection \({\mathbf {i}}_4\) of \({\mathbf {H}}^1(\Omega )\) into \({\mathbf {L}}^4(\Omega )\), we can prove the Lipschitz-continuity of the operator \({\mathbf {T}}_h\) (cf. Lemma 3.7).
Lemma 4.6
Let \(\widetilde{r}\in (0,\widetilde{r}_0)\), with \(\widetilde{r}_0 := \min \{r_1,\widetilde{r}_2\}\), \(r_1\) as in (3.24) and \(\widetilde{r}_2\) as in (4.16). Then, with the same constant \(c_{{\mathbf {T}}_h}>0\) from (4.20), for all \({{\mathbf {w}}}_h, \, \widetilde{{{\mathbf {w}}}}_h\in {\mathbf {W}}_{\widetilde{r}}\) (cf. (4.19)) there holds
We are now in position of establishing the well-posedness of (4.2).
Theorem 4.7
Assume the same hypothesis of Lemma 4.3 and that the data satisfy
Then, the operator \({\mathbf {T}}_h\) has a unique fixed point \({\mathbf {u}}_h\in {\mathbf {W}}_{\widetilde{r}}\) (cf. (4.19)). Equivalently, the coupled problem (4.2) has a unique solution \(({\varvec{\sigma }}_h, {\mathbf {u}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\) and \(({\varvec{\rho }}_{j,h}, \phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\), \(j\in \{1,2\}\), with \({\mathbf {u}}_h\in {\mathbf {W}}_{\widetilde{r}}\). Moreover, there holds
and
Proof
It follows similarly to the proof of Theorem 3.9. Indeed, we first notice from Lemma 4.3 that \({\mathbf {T}}_h\) maps the ball \({\mathbf {W}}_{\widetilde{r}}\) into itself. In turn, it is easy to see from (4.21) (cf. Lemma 4.6) that \({\mathbf {T}}_h : {\mathbf {W}}_{\widetilde{r}}\rightarrow {\mathbf {W}}_{\widetilde{r}}\) is continuous, and hence the existence result follows from the Brouwer fixed-point theorem [14, Theorem 9.9-2]. In addition, it is clear that the estimate (4.24) follows from (4.15), whereas combining (4.9) with (4.15), we obtain (4.23) (cf. (4.20)). On the other hand, the estimate (4.21) and the assumption (4.22) show that \({\mathbf {T}}_h\) is a contraction mapping, which, thanks to the Banach fixed-point theorem, implies the uniqueness result and concludes the proof. \(\square \)
4.3 Specific Finite Element Subspaces
In this section, we introduce specific finite element subspaces satisfying (4.1) and the crucial discrete inf-sup conditions given by hypotheses (H.1) and (H.2). These discrete spaces arise naturally as consequence of the same analysis developed in [11, 25] and [9, Sect. 3] (see also [15, Sect. 5]). Then, with the same notations from Sect. 4.1, for each \(T\in {\mathcal {T}}_h\) we let \(\mathbf {RT}_k(T)\) be the local Raviart–Thomas element of order \(k\ge 0\), i.e., \(\mathbf {RT}_k(T) \,:=\, [\mathrm {P}_k(T)]^n \oplus \mathrm {P}_k(T)\,{\mathbf {x}}\), where \({\mathbf {x}}:= (x_1,\dots , x_n)^\mathrm {t}\) is a generic vector of \(\mathrm {R}^n\) and \(\mathrm {P}_k(T)\) is the space of polynomials defined on T of degree \(\le k\). Then, the finite element subspaces on \(\Omega \) are defined as
It is clear from (4.25) that \(\mathrm {div}({\mathbf {H}}^{{\varvec{\rho }}}_h)\subseteq \mathrm {H}^{\phi }_h\), and hence (4.10) becomes
This fact together with the uniform positiveness of tensors \({\mathbf {Q}}^{-1}_j\) (cf. (2.3)), imply that the bilinear forms \(a_j\) (cf. (2.20)) are \({\mathbf {V}}_{h}\)-elliptic with the same constants \(\alpha _j\) defined in (3.11), and thus the assumption (H.1) is trivially satisfied. In turn, we know from [15, Lemma 5.5] (see also [7, Lemma 4.4] or [9, Lemma 3.3] with \(p=4/3\)) that there holds (H.2).
We end this section by collecting next the approximation properties of the finite element subspaces \({\mathbb {H}}^{{\varvec{\sigma }}}_h\), \({\mathbf {H}}^{{\mathbf {u}}}_h\), \({\mathbf {H}}^{{\varvec{\rho }}}_h\), and \(\mathrm {H}^{\phi }_h\) (cf. (4.25)), whose derivations can be found in [23, 26], and [9, Sect. 3.1] (see also [15, Sect. 5.5]):
\((\mathbf {AP}^{{\varvec{\sigma }}}_h)\) there exists \(C>0\), independent of h, such that for each \(l\in (0,k+1]\), and for each \({\varvec{\tau }}\in {\mathbb {H}}^l(\Omega )\cap {\mathbb {H}}_0(\mathbf {div};\Omega )\) with \(\mathbf {div}\,({\varvec{\tau }}) \in {\mathbf {H}}^l(\Omega )\), there holds
\((\mathbf {AP}^{{\mathbf {u}}}_h)\) there exists \(C>0\), independent of h, such that for each \(l\in [0,k+1]\), and for each \({\mathbf {u}}\in {\mathbf {H}}^{l+1}(\Omega )\), there holds
\((\mathbf {AP}^{{\varvec{\rho }}}_h)\) there exists \(C>0\), independent of h, such that for each \(l\in (0,k+1]\), and for each \({\varvec{\eta }}\in {\mathbf {H}}^{l}(\Omega )\) with \(\mathrm {div}({\varvec{\eta }}) \in \mathrm {W}^{l,4/3}(\Omega )\), there holds
\((\mathbf {AP}^{\phi }_h)\) there exists \(C>0\), independent of h, such that for each \(l\in [0,k+1]\), and for each \(\psi \in \mathrm {W}^{l,4}(\Omega )\), there holds
5 A Priori Error Analysis
In this section, we first derive a Céa estimate for our Galerkin scheme with arbitrary finite element subspaces satisfying the hypothesis stated in Sect. 4.2. Next, using the specific discrete spaces stated in Sect. 4.3, we establish the corresponding rates of convergence. In fact, let \(({\varvec{\sigma }},{\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega ) \times {\mathbf {H}}^1(\Omega )\) and \(({\varvec{\rho }}_j,\phi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\), \(j\in \{1,2\}\), with \({\mathbf {u}}\in {\mathbf {W}}_r\) (cf. (3.37)), be the unique solution of the coupled problem (2.22), and let \(({\varvec{\sigma }}_h, {\mathbf {u}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\) and \(({\varvec{\rho }}_{j,h}, \phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\), \(j\in \{1,2\}\), with \({\mathbf {u}}_h \in {\mathbf {W}}_{\widetilde{r}}\) (cf. (4.19)), be the unique solution of the discrete coupled problem (4.2). Then, we are interested in obtaining an a priori estimate for the error
To this end, we establish next an ad-hoc Strang-type estimate. In what follows, given a subspace \(X_h\) of a generic Banach space \((X,\Vert \cdot \Vert _{X})\), we set for each \(x\in X\)
Lemma 5.1
Let \(\mathrm {H}\) be a reflexive Banach space, and let \(a : \mathrm {H}\times \mathrm {H}\rightarrow \mathrm {R}\) be a bounded bilinear form with induced operator \({\mathcal {A}}\in {\mathcal {L}}(\mathrm {H},\mathrm {H}')\), such that a satisfies the hypotheses of Theorem 3.1. Furthermore, let \(\{\mathrm {H}_h\}_{h>0}\) be a sequence of finite dimensional subspaces of \(\mathrm {H}\), and for each \(h>0\), consider a bounded bilinear form \(a_h : \mathrm {H}_h\times \mathrm {H}_h\rightarrow \mathrm {R}\) with induced operator \({\mathcal {A}}_h\in {\mathcal {L}}(\mathrm {H}_h,\mathrm {H}'_h)\), such that \(a_h|_{\mathrm {H}_h\times \mathrm {H}_h}\) satisfies the hypotheses of Theorem 3.1 as well, with constant \(\widetilde{\alpha }\) independent of h. In turn, given \(F\in \mathrm {H}'\), and a sequence of functionals \(\{F_h\}_{h>0}\), with \(F_h\in \mathrm {H}'_h\) for each \(h>0\), we let \(u\in \mathrm {H}\) and \(u_h\in \mathrm {H}_h \) be the unique solutions, respectively, to the problems
and
Then, there holds
where \(C_{S,1}\) and \(C_{S,2}\) are the positive constants given by
Proof
We proceed similarly to the proof of [33, Theorem 11.1]. Indeed, employing the inf-sup condition of the bilinear form \(a_h\) (cf. (3.1)), the identities (5.1) and (5.2), and the continuity of \({\mathcal {A}}\), \({\mathcal {A}}_h\), F, and \(F_h\), we obtain
where
Then, inspired by [15, Lemma 6.1], we notice that
which implies
Hence, replacing (5.6) back into (5.5) we obtain (5.3) and conclude the proof. \(\square \)
In order to apply Lemma 5.1, we now observe that the problems (2.22) and (4.2) can be rewritten as two pairs of corresponding continuous and discrete formulations, namely
and
where the forms \({\mathcal {A}}_{j,{\mathbf {u}}}\) and \({\mathcal {A}}_{j,{\mathbf {u}}_h}\) are defined as in (3.30).
The following lemma provides a preliminary estimate for the error \(\Vert ({\varvec{\sigma }},{\mathbf {u}}) - ({\varvec{\sigma }}_h,{\mathbf {u}}_h)\Vert \).
Lemma 5.2
There exists a positive constant \(\widetilde{C}_{S,1}\), independent of h, such that
where \(r_1\) and \(C_{{\mathbf {S}}}\) are defined in (3.24) and (3.40), respectively.
Proof
From (3.25) we know that the bounded bilinear forms \(A+B_{\mathbf {u}}\) and \(A+B_{{\mathbf {u}}_h}\) are elliptic with the same constant \(\frac{\alpha _{A}}{2}\). In addition, it is clear that \(F_\mathrm {D}+F_{\underline{\varvec{\phi }}}\) and \(F_\mathrm {D}+F_{\underline{\varvec{\phi }}_h}\) are bounded linear functionals in \({\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) and \({\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\), respectively. Then, applying Lemma 5.1 to the context given by (5.7), we find that
where, according to (5.4), the estimates (3.3), (3.4), and the fact that \({\mathbf {u}}\in {\mathbf {W}}_r, {\mathbf {u}}_h\in {\mathbf {W}}_{\widetilde{r}}\) (cf. (3.37), (4.19)), there holds
Next, using the definition of \(F_{\underline{\varvec{\varphi }}}\) (cf. (2.17)) and its continuity bound (cf. (3.9)), and applying Hölder’s inequality, we readily get
In turn, from the continuity bound of \(B_{{{\mathbf {w}}}}\) (cf. (3.4)) and the fact that \(\big | |{\mathbf {u}}| - |{\mathbf {u}}_h|\big | \le |{\mathbf {u}}- {\mathbf {u}}_h|\), it follows that
Thus, replacing (5.12) and (5.13) back into (5.10), and using the explicit expression of \(\widetilde{C}_{S,2}\) (cf. (5.11)), we find that
which, together with the definition of \(r_1\) and \(C_{{\mathbf {S}}}\) (cf. (3.24) and (3.40)), implies (5.9) and concludes the proof. \(\square \)
Next, we have the following result concerning \(\Vert ({\varvec{\rho }}_j,\phi _j) - ({\varvec{\rho }}_{j,h},\phi _{j,h})\Vert \).
Lemma 5.3
There exists a positive constant \(\widehat{C}_{S,1}\), independent of h, such that
where \(c_{\widetilde{{\mathbf {S}}}}\) is defined at the end of the proof of Lemma 3.3, and \(\widetilde{r}_2\) is given by (4.16).
Proof
We first recall from Sects. 3.3 and 4.2 that the forms \({\mathcal {A}}_{j,{\mathbf {u}}}\) and \({\mathcal {A}}_{j,{\mathbf {u}}_h}\), \(j\in \{1,2\}\), satisfy the hypotheses of Lemma 5.1 on \({\mathbf {H}}:={\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\) and \({\mathbf {H}}_h:={\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\), respectively, the latter with constant \(\widetilde{\alpha } = \frac{\widetilde{\gamma }_j}{2}\) (cf. (4.14)). Therefore, applying Lemma 5.1 to the context (5.8), we arrive at
where, according to (5.4), the definition of the form \({\mathcal {A}}_{j,{{\mathbf {w}}}}\) (cf. (3.30)), the estimates (3.5), (3.6), and (3.7), and the fact that \({\mathbf {u}}\in {\mathbf {W}}_r, {\mathbf {u}}_h\in {\mathbf {W}}_{\widetilde{r}}\) (cf. (3.37), (4.19)), there holds
Next, in order to bound the last term on the right-hand side of (5.15), we notice that the definition of the form \({\mathcal {A}}_{j,{{\mathbf {w}}}}\) (cf. (3.30)) and the continuity bound of \(c_j\) (cf. (3.7)), give
which, together with (5.15), the explicit expression of \(\widehat{C}^j_{S,2}\) (cf. (5.16)) and the bound of \(\Vert \phi _j\Vert _{0,4;\Omega }\) (cf. (3.35)), yields
Then, recalling the definitions of \(c_{\widetilde{{\mathbf {S}}}}\) and \(\widetilde{r}_2\), it is easy to see that (5.17) implies (5.14) with \(\widehat{C}_{S,1} := \max \big \{ \widehat{C}^1_{S,1}, \widehat{C}^2_{S,1} \big \}\), thus concluding the proof. \(\square \)
The required Céa estimate will now follow from (5.9) and (5.14). In fact, we first bound \(\Vert \underline{\varvec{\phi }}- \underline{\varvec{\phi }}_h\Vert _{0,4;\Omega }\) in (5.9) by the right hand side of (5.14). Next, in order to obtain an explicit expression in terms of data, we bound \(\Vert {\mathbf {u}}\Vert _{1,\Omega }\) as in (3.44) instead of directly by r, that is
which, together with the fact that \(C_{{\mathbf {S}}} c_{\widetilde{{\mathbf {S}}}}\) and \(1/r_1, 1/\widetilde{r}_2\) are bounded by \(c_{{\mathbf {T}}}\) and \(1/\widetilde{r}_0\), respectively, allows us to deduce from (5.9) that
Thus, imposing the term that multiplies \(\Vert {\mathbf {u}}- {\mathbf {u}}_h\Vert _{1,\Omega }\) in (5.18) to be sufficiently small, say \(\le 1/2\), we derive the a priori error estimate for \(\Vert ({\varvec{\sigma }},{\mathbf {u}}) - ({\varvec{\sigma }}_h,{\mathbf {u}}_h)\Vert \), which, employed then to bound the last term on the right-hand side of (5.14), provides the corresponding upper bound for \(\sum ^2_{j=1} \Vert ({\varvec{\rho }}_j,\phi _j) - ({\varvec{\rho }}_{j,h},\phi _{j,h})\Vert \). More precisely, we have proved the following result.
Theorem 5.4
Assume that the data \({\mathbf {u}}_{\mathrm {D}}\), \(\underline{\varvec{\phi }}_\mathrm {D}\), and \(\underline{\varvec{\phi }}_{\mathtt {r}}\) satisfy
Then, there exists a positive constant C, independent of h, but depending on \(\nu , \mathtt {F}, \mathtt {R}_j, \Vert {\mathbf {i}}_4\Vert , \Vert {\mathbf {K}}\Vert _{\infty }\), \(\Vert {\mathbf {Q}}^{-1}_j\Vert _{\infty }, \Vert {\mathbf {g}}\Vert _{0,4;\Omega },\alpha _{A}, \gamma _j, \widetilde{\gamma }_j\), \(j\in \{1,2\}\), and the datum \(\underline{\varvec{\phi }}_{\mathrm {D}}\), such that
Now, in order to approximate the pressure, the velocity gradient, and the heat and diffusive vectors, we propose, motivated by (2.7) and (2.9), the expressions
respectively, with \(({\varvec{\sigma }}_h,{\mathbf {u}}_h)\in {\mathbb {H}}^{{\varvec{\sigma }}}_h\times {\mathbf {H}}^{{\mathbf {u}}}_h\) and \(({\varvec{\rho }}_{j,h},\phi _{j,h})\in {\mathbf {H}}^{{\varvec{\rho }}}_h\times \mathrm {H}^{\phi }_h\), \(j\in \{1,2\}\), being the unique solution of the discrete problem (4.2). The corresponding error estimates are established in the following lemma.
Lemma 5.5
Assume that the hypotheses of Theorem 5.4 hold. Let p, \(\nabla {\mathbf {u}}\), and \(\widetilde{{\varvec{\rho }}}_j\), \(j\in \{1,2\}\), be given by (2.7)–(2.9). In addition, let \(p_h\), \(\big (\nabla {\mathbf {u}}\big )_h\), and \(\widetilde{{\varvec{\rho }}}_{j,h}\), \(j\in \{1,2\}\), be the discrete counterparts introduced in (5.20). Then, there exists a positive constant C, independent of h, but depending on \(\nu ,\,\mathtt {F}, \,\mathtt {R}_j, \,\Vert {\mathbf {i}}_4\Vert \), \(\Vert {\mathbf {K}}\Vert _{\infty }, \,\Vert {\mathbf {Q}}^{-1}_j\Vert _{\infty }, \,\Vert {\mathbf {g}}\Vert _{0,4;\Omega }, \,\alpha _{A}, \,\gamma _j, \,\widetilde{\gamma }_j\), \(j\in \{1,2\}\), and the datum \(\underline{\varvec{\phi }}_{\mathrm {D}}\), such that
Proof
It follows from (2.9) and (5.20), adding and subtracting \(\mathtt {R}_j\,\phi _j\,{\mathbf {u}}_h\), that
Next, employing the triangle and Hölder inequalities, and the continuous injection \({\mathbf {i}}_4\) of \({\mathbf {H}}^1(\Omega )\) into \({\mathbf {L}}^4(\Omega )\), we find that
which, together with the estimate (3.35) bounding \(\Vert \phi _j\Vert _{0,4;\Omega }\) and the fact that \({\mathbf {u}}_h\in {\mathbf {W}}_{\widetilde{r}}\) (cf. (4.19)), implies that there exists a positive constant C, depending only on \(\nu , \,\mathtt {F},\, \mathtt {R}_j, \,\Vert {\mathbf {i}}_4\Vert , \,\Vert {\mathbf {K}}\Vert _{\infty }, \,\Vert {\mathbf {Q}}^{-1}_j\Vert _{\infty }, \,\alpha _A, \,\widetilde{\gamma }_j\), \(j\in \{1,2\}\), and the datum \(\underline{\varvec{\phi }}_{\mathrm {D}}\), all of them independent of h, such that
Then, the result is a straightforward consequence of the foregoing inequality and Theorem 5.4. \(\square \)
Finally we complete our a priori error analysis with the corresponding rates of convergence of our Galerkin scheme (4.2), for which we supose in the sequel that the finite element subspaces specified in Sect. 4.3 are employed.
Theorem 5.6
In addition to the hypotheses of Theorems 3.9, 4.7 and 5.4, assume that there exists \(l\in (0,k+1]\) such that \({\varvec{\sigma }}\in {\mathbb {H}}^l(\Omega ) \cap {\mathbb {H}}_0(\mathbf {div};\Omega ),\, \mathbf {div}({\varvec{\sigma }})\in {\mathbf {H}}^l(\Omega ),\, {\mathbf {u}}\in {\mathbf {H}}^{l+1}(\Omega )\), and such that for each \(j\in \{1,2\}\), \({\varvec{\rho }}_j\in {\mathbf {H}}^{l}(\Omega ),\, \mathrm {div}({\varvec{\rho }}_j)\in \mathrm {W}^{l,4/3}(\Omega )\), and \(\phi _j\in \mathrm {W}^{l,4}(\Omega )\). Then, there exists \(C>0\), independent of h, such that
Proof
It follows from the Céa estimate (5.19) and the approximation properties \((\mathbf {AP}^{{\varvec{\sigma }}}_h), (\mathbf {AP}^{{\mathbf {u}}}_h), (\mathbf {AP}^{{\varvec{\rho }}}_h)\) and \((\mathbf {AP}^{\phi }_h)\) specified in Sect. 4.3. \(\square \)
Consequently, from Lemma 5.5 and Theorem 5.6 we obtain the optimal convergence of the post-processed unknowns introduced in (5.20).
Lemma 5.7
Let \(({\varvec{\sigma }},{\mathbf {u}})\in {\mathbb {H}}_0(\mathbf {div};\Omega )\times {\mathbf {H}}^1(\Omega )\) and \(({\varvec{\rho }}_j, \phi _j)\in {\mathbf {H}}(\mathrm {div}_{4/3};\Omega )\times \mathrm {L}^4(\Omega )\), \(j\in \{1,2\}\), be the unique solution of the continuous problem (2.22), and let \(p, \nabla {\mathbf {u}}\), and \(\widetilde{{\varvec{\rho }}}_j\) given by (2.7)–(2.9). In addition, let \(p_h\), \(\big (\nabla {\mathbf {u}}\big )_h\), and \(\widetilde{{\varvec{\rho }}}_{j,h}\) be the discrete counterparts introduced in (5.20). Assume that the hypotheses of Theorem 5.6 hold. Then, there exists \(C>0\), independent of h, such that
6 Numerical Results
In this section we present some examples illustrating the performance of our augmented fully-mixed finite element scheme (4.2), and confirming the rates of convergence provided by Theorem 5.6 and Lemma 5.7. Our implementation is based on a FreeFem++ code [27], in conjunction with the direct linear solver UMFPACK [19]. A Newton–Raphson algorithm with a fixed tolerance \(\textsf {tol}= 1\text {E}-6\) is used for the resolution of the nonlinear problem (4.2). As usual, the iterative method is finished when the relative error between two consecutive iterations of the complete coefficient vector, namely \(\mathbf {coeff}^{m+1}\) and \(\mathbf {coeff}^m\), is sufficiently small, i.e.,
where \(\Vert \cdot \Vert _{\ell ^2}\) is the standard \(\ell ^2\)-norm in \(\mathrm {R}^{\mathtt {DOF}}\) with \(\mathtt {DOF}\) denoting the total number of degrees of freedom generated by the finite element subspaces. The condition of zero-average pressure (translated in terms of the trace of \({\varvec{\sigma }}\)) is imposed through a real Lagrange multiplier.
Errors between exact and approximate solutions are denoted as
In turn, we let \({\mathsf {r}}(\cdot )\) be their corresponding rates of convergence, that is
where h and \(h'\) denote two consecutive meshsizes with errors \({\mathsf {e}}(\diamond )\) and \({\mathsf {e}}'(\diamond )\), respectively.
The examples to be considered in this section are described next. In all of them, for the sake of simplicity, we choose the parameters \(\nu = 1, \varrho = 1, \mathtt {R}_1 = 1, \mathtt {R}_2 = 1\), and \(\underline{\varvec{\phi }}_{\mathtt {r}}=(0,0)\). In turn, and according to (3.36), the stabilization parameters are taken as \(\kappa _1 = \nu \) and \(\kappa _2 = \nu /2\). In addition, in the first two examples the tensors \({\mathbf {K}}, {\mathbf {Q}}_1\), and \({\mathbf {Q}}_2\) are taken as the identity matrix \({\mathbb {I}}\), which satisfy (2.3).
6.1 Example 1: 2D Domain with Different Values of the Parameter \(\mathtt {F}\)
In this first example, we corroborate the rates of convergence in a two dimensional domain and also study the performance of the numerical method with respect to the number of Newton iterations required to achieve certain tolerance when different values of the parameter \(\mathtt {F}\) are given. The domain is the square \(\Omega = (-1, 1)^2\). We consider the potential type gravitational acceleration \({\mathbf {g}}= (0,-1)^\mathrm {t}\), and the data \({\mathbf {f}}(\underline{\varvec{\phi }})\) given in (2.2) is adjusted so that the exact solution is given by the smooth functions
The model problem is then complemented with the appropriate Dirichlet boundary conditions. Tables 1 and 2 show the convergence history for a sequence of quasi-uniform mesh refinements, including the number of Newton iterations when \(\mathtt {F}= 10\). Notice that we are able not only to approximate the original unknowns but also the pressure field, the velocity gradient, and the heat and diffusive vectors through the formulae (5.20). The results confirm that the optimal rates of convergence \(O(h^{k+1})\), provided by Theorem 5.6 and Lemma 5.7 are attained for \(k = 0, 1\). The Newton method exhibits a behavior independent of the meshsize, converging in five iterations in all cases. In Fig. 1 we display the solution obtained with the fully-mixed \({\mathbb {R}}{\mathbb {T}}_1 - {\mathbf {P}}_2 - \mathcal {{\mathbf {R}}}{\mathbf {T}}_1 - \mathrm {P}^{\mathrm {dc}}_1 - \mathcal {{\mathbf {R}}}{\mathbf {T}}_1 - \mathrm {P}^{\mathrm {dc}}_1\) approximation with \(1,176,134\,\mathtt {DOF}\), where \(\mathrm {P}^{\mathrm {dc}}_1\) denotes the piecewise linear discontinuous finite element.
Example 1, Computed magnitude of the pseudostress tensor component, velocity, and pseudoheat vector, and temperature field (top plots); magnitude of the pseudodiffusive vector, concentration field, magnitude of the heat vector and pressure field (bottom plots)
On the other hand, in Table 3 we show the behaviour of the iterative method with polynomial degree \(k=0\), as a function of the parameter \(\mathtt {F}\in \{10^0, 10^1, 10^2, 10^3, 10^4, 10^5\}\), considering different meshsizes h, and a tolerance \(\textsf {tol}= 1\text {E}-06\). Here we observe that the higher the parameter \(\mathtt {F}\) the higher the number of Newton iterations required.
6.2 Example 2: Convergence Against Smooth Exact Solutions in a 3D Domain
In our second example, we consider the cube domain \(\Omega = (0, 1)^3\). The solution is given by
Similarly to the first example, we consider \(\mathtt {F}= 10\) and the potential type gravitational acceleration \({\mathbf {g}}= (0,0,-1)^\mathrm {t}\), whereas the data \({\mathbf {f}}(\underline{\varvec{\phi }})\) is computed from (2.2) using the above solution. The numerical solutions are shown in Fig. 2, which were built using the fully-mixed \(\mathbb {RT}_0 - {\mathbf {P}}_1 - \mathbf {RT}_0 - \mathrm {P}_0 - \mathbf {RT}_0 - \mathrm {P}_0\) approximation with \(2,497,827\,\mathtt {DOF}\). The convergence history for a set of quasi-uniform mesh refinements using \(k=0\) is shown in Table 4. Again, the mixed finite element method converges optimally with order \({\mathcal {O}}(h)\), as it was proved by Theorem 5.6 and Lemma 5.7.
Example 2, Computed magnitude of the pseudostress tensor component, velocity, and pseudoheat vector, and temperature field (top plots); magnitude of the pseudodiffusive vector, concentration field, magnitude of the diffusive vector and pressure field (bottom plots)
6.3 Example 3: Flow Through Porous Media with Channel Network
In our last example, inspired by [2, Sect. 5.2.4], we focus on flow through porous media with channel network. We consider the square domain \(\Omega = (-1,1)^2\) with an internal channel network denoted as \(\Omega _{\mathrm {c}}\), which is described in the first plot of Fig. 3. We consider the coupling of the Brinkman–Forchheimer and double-diffusion equations (2.8) in the whole domain \(\Omega \) with \({\mathbf {Q}}_1 = 0.5\,{\mathbb {I}}, {\mathbf {Q}}_2 = 0.125\,{\mathbb {I}}\), but with different values of the parameters \(\mathtt {F}\) and \({\mathbf {K}}= \alpha \,{\mathbb {I}}\) for the interior and the exterior of the channel, that is,
The parameter choice corresponds to increased inertial effect (\(\mathtt {F}= 10\)) in the channel and a high permeability (\(\alpha = 1\)), compared to reduced inertial effect (\(\mathtt {F}= 1\)) in the porous media and low permeability (\(\alpha = 0.001\)). In addition, the boundaries conditions are
In particular, the first row of boundary equations corresponds to inflow on the left boundary and zero stress outflow on the rest of the boundary. We stress here that, using similar arguments to those employed in [12], we are able to extended our analysis to the present case of mixed boundary conditions for the double-diffusion equations. In Fig. 3 we display the computed magnitude of the pseudostress tensor component, velocity, velocity gradient component, pseudoheat and pseudodiffusive vectors, and the temperature and concentration fields, which were built using the fully-mixed \(\mathbb {RT}_0 - {\mathbf {P}}_1 - \mathbf {RT}_0 - \mathrm {P}_0 - \mathbf {RT}_0 - \mathrm {P}_0\) approximation on a mesh with 27, 287 triangle elements (actually representing \(257,284\,\mathtt {DOF}\)). As expected, we observe faster flow through the channel network, with a significant velocity gradient across the interface between the channel and the porous media. The pseudostress is more diffused, since it includes the pressure field. The temperature is higher on the left of the boundary and goes decaying to the right of the domain. Notice that both the pseudoheat and pseudodiffusive vectors are higher in the channel. This example illustrates the ability of the coupling of the Brinkman–Forchheimer and double-diffusion equations to handle heterogeneous media using spatially varying parameters. The example is particularly challenging, due to the strong jump discontinuity of the parameters across the two regions, which are handled very well by our numerical method.
Example 3, Domain configuration, computed magnitude of the pseudostress tensor component, velocity, and velocity gradient component (top plots); magnitude of the pseudoheat vector, temperature field, magnitude of the pseudodiffusive vector, and concentration field (bottom plots)
References
Alzahrani, A.K.: Importance of Darcy–Forchheimer porous medium in \(3\)D convective flow of carbon nanotubes. Phys. Lett. A 382(40), 2938–2943 (2018)
Ambartsumyan, I., Khattatov, E., Nguyen, T., Yotov, I.: Flow and transport in fractured poroelastic media. GEM Int. J. Geomath. 10(1), 34 (2019)
Bhatti, M.M., Zeeshan, A., Ellahi, R., Shit, G.C.: Mathematical modeling of heat and mass transfer effects on MHD peristaltic propulsion of two-phase flow through a Darcy–Brinkman–Forchheimer porous medium. Adv. Powder Technol. 29(5), 1189–1197 (2018)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)
Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27–34 (1949)
Bürger, R., Méndez, P.E., Ruiz-Baier, R.: On \({\mathbf{H}}(\div )\)-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal. 57(3), 1318–1343 (2019)
Camaño, J., García, C., Oyarzúa, R.: Analysis of a conservative mixed-FEM for the stationary Navier–Stokes problem. Preprint 2018-25, Centro de Investigación en Ingeniería Matemática (\(\text{CI}^2\)MA), Universidad de Concepción, Chile (2018)
Camaño, J., Gatica, G.N., Oyarzúa, R., Tierra, G.: An augmented mixed finite element method for the Navier–Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54(2), 1069–1092 (2016)
Camaño, J., Muñoz, C., Oyarzúa, R.: Numerical analysis of a dual-mixed problem in non-standard Banach spaces. Electron. Trans. Numer. Anal. 48, 114–130 (2018)
Caucao, S., Gatica, G.N., Ortega, J.P.: A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations. (in preparation)
Caucao, S., Gatica, G.N., Oyarzúa, R.: Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations. ESAIM Math. Model. Numer. Anal. 52(5), 1947–1980 (2018)
Caucao, S., Oyarzúa, R., Villa-Fuentes, S.: A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy. Calcolo (to appear)
Celebi, A.O., Kalantarov, V.K., Ugurlu, D.: On continuous dependence on coefficients of the Brinkman–Forchheimer equations. Appl. Math. Lett. 19(8), 801–807 (2006)
Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia (2013)
Colmenares, E., Gatica, G.N., Moraga, S.: A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem. ESAIM Math. Model. Numer. Anal. 54(5), 1525–1568 (2020)
Colmenares, E., Gatica, G.N., Moraga, S., Ruiz-Baier, R.: A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system. SMAI J. Comput. Math. 6, 125–157 (2020)
Colmenares, E., Gatica, G.N., Oyarzúa, R.: An augmented fully-mixed finite element method for the stationary Boussinesq problem. Calcolo 54(1), 167–205 (2017)
Colmenares, E., Gatica, G.N., Oyarzúa, R.: Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem. C. R. Math. Acad. Sci. Paris 354(1), 57–62 (2016)
Davis, T.A.: Algorithm 832: UMFPACK V43—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196–199 (2004)
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)
Faulkner, J., Hu, B.X., Kish, S., Hua, F.: Laboratory analog and numerical study of ground water flow and solute transport in a karst aquifer with conduit and matrix domains. J. Contam. Hydrol. 110(1–2), 34–44 (2009)
Forchheimer, P.: Wasserbewegung durch boden. Z. Ver. Deutsch Ing. 45, 1782–1788 (1901)
Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014)
Gatica, G.N., Gatica, L.F., Márquez, A.: Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126(4), 635–677 (2014)
Gatica, L.F., Oyarzúa, R., Sánchez, N.: A priori and a posteriori error analysis of an augmented mixed-FEM for the Navier–Stokes–Brinkman problem. Comput. Math. Appl. 75(7), 2420–2444 (2018)
Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)
Kaloni, P.N., Guo, J.: Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman–Forchheimer model. J. Math. Anal. Appl. 204(1), 138–155 (1996)
Kufner, A., Jhon, O., Fučík, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden (1977)
Ôtani, M., Uchida, S.: Global solvability of some double-diffusive convection system coupled with Brinkman–Forchheimer equations. Lib. Math. (N.S.) 33(1), 79–107 (2013)
Payne, L.E., Straughan, B.: Convergence and continuous dependence for the Brinkman–Forchheimer equations. Stud. Appl. Math. 102(4), 419–439 (1999)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)
Roberts, J.E., Thomas, J.M.: Mixed and Hybrid Methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam (1991)
Safi, S., Benissaad, S.: Double-diffusive convection in an anisotropic porous layer using the Darcy–Brinkman–Forchheimer formulation. Arch. Mech. (Arch. Mech. Stos.) 70(1), 89–102 (2018)
Zhuang, Y.J., Yu, H.Z., Zhu, Q.Y.: A thermal non-equilibrium model for \(3\)D double diffusive convection of power-law fluids with chemical reaction in the porous medium. Int. J. Heat Mass Transf. 115–B, 670–694 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was partially supported by CONICYT-Chile through the Project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, the Project PAI77190084 of the PAI Program: Convocatoria Nacional Subvención a la Instalación en la Academia, and Fondecyt Project 11121347; by Centro de Investigación en Ingeniería Matemática (\(\hbox {CI}^2\)MA), Universidad de Concepción; and by Universidad del Bío-Bío through DIUBB Project 151408 GI/VC.
Rights and permissions
About this article
Cite this article
Caucao, S., Gatica, G.N., Oyarzúa, R. et al. A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations. J Sci Comput 85, 44 (2020). https://doi.org/10.1007/s10915-020-01305-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01305-x
Keywords
- Brinkman–Forchheimer equations
- Double-diffusive convection system
- Stress–velocity formulation
- Fixed point theory
- Mixed finite element methods
- A priori error analysis