Abstract
We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fully-mixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68–95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.
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1 Introduction
We have recently introduced a mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium described by Brinkman equations, and its interaction with pure porous media flow under Darcy’s law [1]. There, the system is formulated in terms of velocity and pressure of the non-viscous flow, together with vorticity, velocity and pressure of the Brinkman region. The tangential vorticity vanishes on the boundary of the Brinkman domain, whereas slip velocity conditions are assumed on the overall boundary. The corresponding mixed variational formulation leads to a Lagrange multiplier enforcing pressure continuity across the interface, while mass balance results from essential boundary conditions on each domain. As a consequence, a classical saddle-point operator equation is obtained, whose invertibility hinges on the well-known Babuška–Brezzi theory. A similar treatment is used to establish the solvability of the discrete problem associated to the Galerkin method. The needed continuous and discrete inf-sup conditions can be guaranteed thanks to the so-called T-coercivity argument (cf. [18, 26] and the references therein), where one defines adequate injective operators delivering lower bounds of the corresponding suprema. As the stability of the Galerkin scheme requires that the \({\mathbf {curl}}\) of the discrete vorticity space is contained in the discrete Brinkman velocity space, we specify Raviart–Thomas and Nédélec spaces for the approximation of the global velocity and the Brinkman vorticity, respectively.
On the other hand, the derivation of adaptive schemes for transmission free flow—porous media problems has been extensively studied in recent years. In particular, we refer to [6, 9,10,11, 14, 16, 27], which are focused on Stokes–Darcy and Navier–Stokes/Darcy couplings, and where the interface conditions are treated in different ways, from both mathematical and numerical perspective. For instance, in [6, 14, 16, 27], Beavers–Joseph–Saffman-type conditions are considered on the interface, whereas in [10, 11], similar transmission conditions to those employed in [1] are assumed. Also, an interesting feature of the proof of reliability in [6], which differs from the approaches in the other works, is the utilisation of intermediate inf-sup inequalities that are obtained along the proof of the global inf-sup condition. Differently from the above, and similarly as in [14, 16, 27], the efficiency estimates in [6] follow from usual arguments based on inverse inequalities and the localisation technique employing triangle-bubble and edge-bubble functions. In turn, the assumption of a smallness condition on the data is the distinctive feature of the approach in [16], where a reliable and efficient residual-based a posteriori error estimator for the three dimensional version of the augmented-mixed method introduced in [17], is derived. Furthermore, an a posteriori error estimator for a conforming and nonconforming vorticity-based finite element method of a Stokes–Darcy coupled problem was derived in [10, 11], respectively, but the resulting estimate in [11] is not optimal. In addition, even though in [10, 11] the model problem is addressed for both two and three spatial dimensions, the corresponding a posteriori error analysis is explicitly derived only for the 2D case.
According to the previous discussion, and as a natural continuation of the a priori error analysis developed in [1], our goal in the present paper is to provide a reliable and efficient residual-based a posteriori error estimator for the finite element method introduced and analyzed in that reference. In this way, we aim to improve the accuracy of the discrete scheme from [1] in different scenarios, including presence of singularities or high gradients of the solution. Indeed, in contrast with the methodology developed in [10, 11], and following the approaches in [16, 27], we highlight that the derivation of our error estimator is based on a global inf-sup condition in combination with suitable Helmholtz decompositions adapted from [16, 31], and local approximation properties of Clément, Raviart–Thomas, and Clément-type Nédélec interpolators. Then, similarly as in [14, 16, 27], the associated efficiency estimates are consequence of suitable inverse inequalities and local bounds for tetrahedron-bubble and facet-bubble functions.
The remainder of the paper is structured in the following manner. General preliminary notation is presented in the last part of this section. The model problem and the mixed variational formulation are outlined in Sect. 2, where we also recall its unique solvability and the mixed finite element discretisation. The core of the present analysis is contained in Sect. 3, where we define the error estimator and provide a detailed derivation of its reliability and efficiency. Finally, Sect. 4 gives two numerical tests aimed to illustrate the features of the method and the proposed estimator.
Some recurrent notation to be employed throughout the paper includes the following. If \(S \subseteq \mathbb {R}^3\) is a domain or a Lipschitz surface, and \(r\in \mathbb {R}\), we set vectorial Sobolev spaces as \({\mathbf {H}}^r(S):= [\mathrm {H}^r(S)]^3\), adopt the convention \({\mathbf {H}}^0(S) \equiv {\mathbf {L}}^2(S)\), and denote the corresponding norms by \(\Vert \cdot \Vert _{r,S}\) (for both \(\mathrm {H}^r(S)\) and \({\mathbf {H}}^r(S)\)). In general, given a generic Hilbert space \(\mathrm {H}\), we will employ \({\mathbf {H}}\) to denote its vectorial counterpart \(\mathrm {H}^3\). We also recall the definition of the Hilbert spaces
normed, respectively, with
where, for any vector field \(\varvec{v}:=(v_1,v_2,v_3)^\mathtt{t} \in {\mathbf {L}}^2(S)\) we have
In addition we will use the space
endowed with the the usual norm of \(\mathrm {L}^2(S)\). In turn, for each integer \(k\ge 0\) we denote by \(P_k(S)\) the space of polynomials in S of total degree \(\le k\), and we set \({\mathbf {P}}_k(S) = [P_k(S)]^3\). Finally, the symbol \({\mathbf {0}}\) will stand for a generic null vector (including the null functional and operator), and C (indistinguishably c, with or without subscripts, bars, tildes or hats) will denote generic constants independent of the discretisation parameters.
2 Governing equations and a mixed variational formulation
2.1 The continuous model
We first let \({\Omega _{\mathrm {B}}}\) and \({\Omega _{\mathrm {D}}}\) be bounded and simply connected polyhedral Lipschitz domains in \(\mathbb {R}^3\) such that \(\partial {\Omega _{\mathrm {B}}}\cap \partial {\Omega _{\mathrm {D}}}=: \Sigma \not = \emptyset \) and \({\Omega _{\mathrm {B}}}\cap {\Omega _{\mathrm {D}}}= \emptyset \), and set \(\Omega := {\Omega _{\mathrm {B}}}\cup \Sigma \cup {\Omega _{\mathrm {D}}}\) with boundary \(\Gamma =\partial \Omega \) split into \({\Gamma _{\mathrm {B}}}\subseteq \partial {\Omega _{\mathrm {B}}}\) and \({\Gamma _{\mathrm {D}}}\subseteq \partial {\Omega _{\mathrm {D}}}\). Then, given source terms \(\varvec{f}_{\mathrm {D}}\in \mathbf {L}^2({\Omega _{\mathrm {D}}})\) and \(\varvec{f}_{\mathrm {B}}\in \mathbf {L}^2({\Omega _{\mathrm {B}}})\), we are interested in the Brinkman–Darcy coupled problem
which is formulated in terms of the Brinkman velocity \(\varvec{u}_\mathrm {B}\), the Brinkman pressure \(p_\mathrm {B}\), the Brinkman vorticity \(\varvec{\omega }_\mathrm {B}\), the Darcy velocity \(\varvec{u}_\mathrm {D}\), and the Darcy pressure \(p_\mathrm {D}\). Here \(\varvec{n}\) stands for the outward normal at \({\Omega _{\mathrm {B}}}\) and \({\Omega _{\mathrm {D}}}\), \(\nu >0\) is the kinematic viscosity of the fluid, and \(\kappa _{\mathrm {D}},\kappa _{\mathrm {B}}>0\) are the absolute permeabilities of the Darcy and Brinkman subdomains, respectively.
The boundary conditions on the Brinkman and Darcy subdomains suggest the following spaces
In addition, the pressure continuity across the interface \(\Sigma \) allows us to define its trace via the auxiliary unknown \( \lambda := p_\mathrm {D}|_{\Sigma } = p_\mathrm {B}|_{\Sigma } \in \mathrm {H}^{1/2}(\Sigma ), \) where \(\langle \cdot ,\cdot \rangle _\Sigma \) denotes the duality pairing of \(\mathrm {H}^{-1/2}(\Sigma )\) and \(\mathrm {H}^{1/2}(\Sigma )\) with respect to the \(\mathrm {L}^2(\Sigma )\)-inner product. In turn, the continuity of normal velocities across \(\Sigma \) is imposed in a weak manner as
Then, a fully-mixed formulation for (2.1) reads as follows: Find \(\mathbf {\varvec{u}} \!:=\! (\varvec{u}_\mathrm {B},\varvec{\omega }_\mathrm {B},\varvec{u}_\mathrm {D})\!\in {\mathbf {H}}\) and \(\mathbf {p} :=(p_\mathrm {B},p_\mathrm {D},\lambda ) \in {\mathbf {Q} }_{0}\) such that
where \({\mathbf {H}} := {\mathbf {H}}_{\mathrm B}({\mathrm {div}};{\Omega _{\mathrm {B}}}) \times \mathbf {H}_0({\mathbf {curl}};{\Omega _{\mathrm {B}}}) \times \mathbf {H}_{\mathrm D}(\mathop {\mathrm {div}}\nolimits ;{\Omega _{\mathrm {D}}})\), \({\mathbf {Q}}_{0} := \mathrm {L}^2_{0}({\Omega _{\mathrm {B}}}) \times \mathrm {L}^2({\Omega _{\mathrm {D}}}) \times \mathrm H^{1/2}(\Sigma )\), and the bilinear forms \({\mathbf {a}} : {\mathbf {H}} \times {\mathbf {H}} \rightarrow \mathbb {R}\) and \({\mathbf {b}} : {\mathbf {H}} \times {\mathbf {Q}}_{0} \rightarrow \mathbb {R}\), and the functional \({\mathbf {F}} \in {\mathbf {H}}'\), are defined by
and
for all \(\mathbf {\varvec{u}} := (\varvec{u}_\mathrm {B},\varvec{\omega }_\mathrm {B},\varvec{u}_\mathrm {D}), \,\, \mathbf {\varvec{v}} := (\varvec{v}_\mathrm {B},\varvec{z}_\mathrm {B},\varvec{v}_\mathrm {D}) \in {\mathbf {H}}\), and for all \(\mathbf {q} :=(q_\mathrm {B},q_\mathrm {D},\xi ) \in {\mathbf {Q}}_{0}\).
The well-posedness of (2.2) has been established in [1] using the classical Babuška–Brezzi theory:
Theorem 1
There exists a unique \(({\mathbf {\varvec{u}}},\mathbf {p}):=\) \(\big ((\varvec{u}_\mathrm {B},\varvec{\omega }_\mathrm {B},\varvec{u}_\mathrm {D}),(p_\mathrm {B},p_\mathrm {D},\lambda ) \big ) \in {\mathbf {H}} \times {\mathbf {Q}}_0\) solution of the mixed formulation (2.2). Moreover, there exists \(c > 0\) such that
2.2 Discretisation using a finite element method
Let \(\mathcal {T}_{h}^{{\mathrm B}}\) and \(\mathcal {T}_{h}^{{\mathrm D}}\) be respective partitions of \(\Omega _{\mathrm B}\) and \(\Omega _{\mathrm D}\) by shape-regular tetrahedra T of diameter \(h_T\). We assume that these tetrahedrisations match on the interface so that \(\mathcal {T}_{h}:=\mathcal {T}_{h}^{{\mathrm B}}\cup \mathcal {T}_{h}^{{\mathrm D}}\) is a regular family of triangulations of \(\Omega ={\Omega _{\mathrm {B}}}\cup \Sigma \cup {\Omega _{\mathrm {D}}}\), with meshsize \(h:=\max \{h_T:\; T\in \mathcal {T}_{h}\}\). We denote by \(\Sigma _{h}\) the triangulation on \(\Sigma \) induced by \(\mathcal {T}_{h}\), which is formed by triangles F of diameter \(h_{F}\), and set \(h_{\Sigma }:=\max \{h_F:\; F\in \Sigma _{h}\}\). Next we introduce the finite-dimensional spaces
where \(\star \in \{{\mathrm B},{\mathrm D}\}\), and for any \(T\in \mathcal {T}_{h}^{\star }\) we denote by \({\mathbb {RT}_0}(T):=\mathbf {P}_{0}(T)\oplus P_{0}(T)\,\varvec{x}\) the local Raviart–Thomas space of lowest order. In addition, we set
where for any \(T\,\in \,\mathcal {T}_{h}^{{\mathrm B}}\), \({\mathbb {ND}_1}(T):=\mathbf {P}_{0}(T)\oplus \mathbf {P}_{0}(T)\times \varvec{x}\) is the local edge space of Nédélec type
The approximation of the interface unknown will occur on an independent triangulation \(\widetilde{\Sigma }_{h}\) of \(\Sigma \), by elements \(\widetilde{F}\) of maximum diameter \(h_{\widetilde{\Sigma }}:=\max \left\{ h_{\widetilde{F}}:\; \widetilde{F}\in \widetilde{\Sigma }_{h}\right\} \), where we define the space
In this way the Galerkin scheme associated to (2.2) reads: Find \(\mathbf {\varvec{u}}_h := (\varvec{u}_h^\mathrm {B},\varvec{\omega }_h^\mathrm {B},\varvec{u}_h^\mathrm {D})\in {\mathbf {H}}_h\) and \(\mathbf {p}_h :=(p_h^\mathrm {B},p_h^\mathrm {D},\lambda _h) \in {\mathbf {Q}}_{0,h}\) such that
where \({{\mathbf {H}}}_h := {{\mathbf {H}}}^{\mathrm B}_h \times {{\mathbf {H}}}^{\mathrm B}_{0,h} \times {{\mathbf {H}}}^{\mathrm D}_h\) and \({\mathbf {Q}}_{0,h} := \mathrm {Q}^{\mathrm B}_{h,0} \times \mathrm {Q}^{\mathrm D}_h \times \mathrm {Q}^\Sigma _{h}\). We point out that the solvability of (2.9) requires the mesh condition \(h_{\Sigma }\le C_{0}\,h_{\widetilde{\Sigma }}\), where \(C_{0}\) is a positive constant. Details are to be found in [1, §4.2.3-4.2.4].
3 A residual-based a posteriori error estimator
In this section we derive a reliable and efficient a posteriori error estimator for the Galerkin scheme (2.9). Most of the present proofs make extensive use of estimates available in [1, 3, 5, 6, 8, 15, 21, 23, 24, 27].
3.1 Preliminaries
Given a tetrahedron \(T\in \mathcal {T}_{h}\), we let \(\mathcal {E}(T)\) and \(\mathcal {F}(T)\) be the sets of its edges and faces, respectively. In addition, we denote by \(\mathcal {E}_{h}\) and \(\mathcal {F}_{h}\) be the sets of all edges and faces of \(\mathcal {T}_{h}\), respectively, so that \(\mathcal {F}_{h}\) is subdivided as follows:
where \(\mathcal {F}_{h}(\Gamma _{\star }):=\{F\in \mathcal {F}_{h}:\; F\subseteq \Gamma _{\star }\}\), \(\mathcal {F}_{h}(\Omega _{\star }):=\{F\in \mathcal {F}_{h}:\; F\subseteq \Omega _{\star }\}\), for each \(\star \in \{{\mathrm B},{\mathrm D}\}\), and \(\mathcal {F}_{h}(\Sigma ):=\{F\in \mathcal {F}_{h}:\; F\subseteq \Sigma \}\). In turn, for each \(T\in \mathcal {T}_{h}\) we denote
and \(\mathcal {F}_{h,T}(\Sigma ):=\{F\in \partial T: \,\,F\in \mathcal {F}_{h}(\Sigma )\}\). Also, for each face \(F\in \mathcal {F}_{h}(\Omega _{\star })\) we fix a unit normal \(\varvec{n}_{F}\) to F, so that given \(\varvec{v}\in \mathbf {L}^{2}(\Omega _{\star })\) such that \(\varvec{v}|_{T}\in \mathbf {C}(T)\) on each \(T\in \mathcal {T}_{h}^{\star }\), and given \(F\in \mathcal {F}_{h}(\Omega _{\star })\), we let \(\llbracket \varvec{v}\times \varvec{n}_{F}\rrbracket \) be the corresponding jump of the tangential traces across F, that is \(\llbracket \varvec{v}\times \varvec{n}_{F}\rrbracket :=(\varvec{v}|_{T}-\varvec{v}|_{T'})|_{F}\times \varvec{n}_{F}\), where T and \(T'\) are the elements of \(\mathcal {T}_{h}^{\star }\) having F as a common face. In addition, for each edge E of a tetrahedron \(T\in \mathcal {T}^{\star }_{h}\), we fix a unit tangential vector \(\varvec{t}_{E}\) along E. When no confusion arises, we simple write \(\varvec{n}\) instead of \(\varvec{n}_{F}\), and \(\varvec{t}\) instead \(\varvec{t}_{E}\).
We now recall from [13] the tangential \({\mathbf {curl}}\) operator \({\mathbf {curl}}_{s}:\mathrm {H}^{1/2}(\Sigma )\rightarrow \mathcal {L}(\mathrm {H}^{-1/2}(\Sigma ))\), with \(\mathcal {L}(\mathrm {H}^{-1/2}(\Sigma ))\) denoting the tangential vector fields of order \(-1/2\), which is defined by \({\mathbf {curl}}_{s}(\chi ):=\nabla \chi \times \varvec{n}\), for any sufficiently smooth function \(\chi \). This is a linear and continuous map (see [13, Propositions 3.4 and 3.6]) which will be required in the sequel. We will also make use of the Raviart–Thomas interpolator of lowest order (see [22]) \(\Pi _{h}^{\star }:\mathbf {H}^{1}(\Omega _{\star })\rightarrow \mathbf {H}_{h}^{\star }\), \(\star \in \{{\mathrm B},{\mathrm D}\}\), which according to its characterisation given by the identity
verifies that
where \(\mathcal {P}_{h}^{\star }\) is the \(\mathrm {L}^{2}(\Omega _{\star })\)-orthogonal projector onto \(\mathrm {P}_{0}(\Omega _{\star })\). In addition, we recall the Clément operator onto the space of the continuous piecewise linear functions \(\,\mathrm {I}^{\star }_{h}:\mathrm {H}^{1}(\Omega _{\star })\rightarrow \mathrm {X}^{\star }_{h}\) (cf. [20]), where
and let \(\mathbf {I}_{h}^{\star }:\mathbf {H}^{1}(\Omega _{\star })\rightarrow \mathbf {X}_{h}^{\star }\) be its vectorial counterpart defined component-wise. These maps satisfy the following properties (see [12, 20, 22], respectively)
Lemma 1
There exist \(c_1, c_2>0\), independent of h, such that for all \(\varvec{v}\in \mathbf {H}^{1}(\Omega _{\star })\) there hold
where \(T_{F}\) is a tetrahedron of \(\mathcal {T}_{h}^{\star }\) containing a face F on its boundary.
Lemma 2
There exist constants \(c_{3},c_{4}>0\), independent of h, such that for all \(v\in \mathrm {H}^{1}(\Omega _{\star })\) there hold
where
Furthermore, following [21] we define the Clément-type Nédélec interpolator \(\varvec{\mathcal {N}}_{h}: \mathbf {L}^{2}({\Omega _{\mathrm {B}}})\rightarrow \mathbf {H}_{h,0}^{{\mathrm B}}\) by:
where \(\mathcal {E}_{h}({\Omega _{\mathrm {B}}})\) is the set of interior edges of \(\mathcal {T}_{h}^{{\mathrm B}}\), \(\Delta _{{\mathrm B}}(E):=\cup \{T'\in \mathcal {T}^{{\mathrm B}}_{h}:\quad T'\cap E\ne \emptyset \}\), and \(\lambda _{E}\) is the standard basis function for the lowest order Nédelec element, which satisfies
where \(\delta _{E,E'}\) is the Kronecker delta. The approximation properties of \(\varvec{\mathcal {N}}_{h}\) are summarised in the following Lemma (see [21, §4.3, Theorem 4.2, and §6] and also [8, Proposition 2]).
Lemma 3
There exist \(c_{5},c_{6}>0\), independent of h, such that for all \(\varvec{\psi }\in \mathbf {H}_{0}({\mathbf {curl}},{\Omega _{\mathrm {B}}})\cap \mathbf {H}^{1}({\Omega _{\mathrm {B}}})\),
We will also require stable Helmholtz decompositions for \(\mathbf {H}_{\star }({\mathrm {div}};\Omega _{\star })\) with \(*\in \{{\mathrm B},{\mathrm D}\}\). A technical assumption is that \(\Gamma _{\star }\) lies on the “convex part” of \(\Omega _{\star }\), signifying that there exists a convex domain containing \(\Omega _{\star }\), whose boundary contains \(\Gamma _{\star }\). More precisely, introducing the space
we have the following result shown in [23, Theorem 3.2].
Lemma 4
Assume that there exists a convex domain \(\Xi _{\star }\) such that \(\Omega _{\star }\subseteq \Xi _{\star }\) and \(\Gamma _{\star }\subseteq \partial \Xi _{\star }\). Then, given \(\varvec{v}_{\star }\in \mathbf {H}_{\star }({\mathrm {div}};\Omega _{\star })\) there exist \(w\in \mathrm {H}^{2}(\Omega _{\star })\) and \(\varvec{\beta }\in \mathbf {H}^{1}_{\Gamma _{\star }}(\Omega _{\star })\) such that
where \(C_{\star }\) is a positive constant independent of all the foregoing variables.
In turn, a decomposition for \(\mathbf {H}_{0}({\mathbf {curl}};{\Omega _{\mathrm {B}}})\) is given as follows.
Lemma 5
Given \(\varvec{z}_\mathrm {B}\in \mathbf {H}_{0}({\mathbf {curl}};{\Omega _{\mathrm {B}}})\) there exist \(\varvec{\varphi }\in \mathbf {H}^{1}_{0}({\Omega _{\mathrm {B}}})\), \(\chi \in \mathrm {H}_{0}^{1}({\Omega _{\mathrm {B}}})\), and \(C>0\) such that
and
Proof
See [31, Lemma 2.2 and §5] \(\square \)
We end this section with an estimate (in terms of local quantities) for the \(\mathrm {H}^{-1/2}(\Sigma )\) norm of functions in a particular subspace of \(\mathrm {H}^{-1/2}(\Sigma )\cap \mathrm {L}^{2}(\Sigma )\). According to the definition of \(\mathrm {Q}_{h}^{\Sigma }\) (cf. 2.8), we introduce the following orthogonal-type space
Lemma 6
Assume that for each \(F\in \Sigma _{h}\) there exists \(\widetilde{F}\in \widetilde{\Sigma }_{h}\) such that \(F\subseteq \widetilde{F}\) and \(h_{\Sigma }\,\le \,C_{1}\,h_{\widetilde{\Sigma }}\), with a constant \(C_{1}>0\) independent of \(h_{\Sigma }\) and \(h_{\widetilde{\Sigma }}\). Then, there exists \(C>0\) independent of the aforementioned meshsizes, such that
Proof
See [16, Lemma 3.4]. \(\square \)
3.2 Defining the proposed estimator
Given \((\mathbf {{\varvec{u}}}_{h},\mathbf {p}_{h}):=\big ((\varvec{u}_h^\mathrm {B},\varvec{\omega }_h^\mathrm {B},\varvec{u}_h^\mathrm {D}),(p_h^\mathrm {B},p_h^\mathrm {D},\lambda _h) \big ) \in {{\mathbf {H}}}_{h}\times \mathbf {Q}_{h,0}\) the unique solution of (2.9), we define for each \(T\in \mathcal {T}_{h}^{{\mathrm B}}\), the local a posteriori error indicator \(\Theta _{{\mathrm B},T}\) as follows:
and for each \(T\in \mathcal {T}_{h}^{{\mathrm D}}\), we define the local a posteriori error indicator \(\Theta _{{\mathrm D},T}\) as
It is not difficult to see that each term defining \(\Theta _{{\mathrm B},T}^{2}\) and \(\Theta _{{\mathrm D},T}^{2}\) is residual. Hence a global residual error estimator for (2.9) can be defined as
The remainder of this section advocates to establish the existence of positive constants \(C_\mathtt{{eff}}\) and \(C_\mathtt{{rel}}\), independent of the meshsizes and the continuous and discrete solutions, such that
where \(\mathrm{h.o.t}\) stands, eventually, for one or several terms of higher order. The upper and lower bounds in (3.8), are derived below in Sects. 3.3 and 3.4, respectively.
3.3 Reliability
3.3.1 Preliminary estimates
We begin by recalling that the first inequality in the continuous dependence result (2.3) is equivalent to the global inf-sup condition
for all \((\mathbf {\varvec{w}},\mathbf {r})\in {\mathbf {H}}\times {\mathbf {Q}}_{0}\). This allows to establish a first estimate for the total error as follows.
Theorem 2
Let \((\mathbf {{\varvec{u}}},\mathbf {p})\in {\mathbf {H}} \times {\mathbf {Q}}_0\) and \((\mathbf {{\varvec{u}}}_{h},\mathbf {p}_{h})\in {{\mathbf {H}}}_{h} \times {\mathbf {Q}}_{0,h}\) be the unique solutions of (2.2) and (2.9), respectively. Then, there exists a constant \(C>0\), independent of h, such that
where \({\mathbf {E}}\in {\mathbf {H}}'\) is defined by
and satisfies
Proof
Applying (3.9) to the error \((\mathbf {\varvec{w}},\mathbf {r}):=(\mathbf {{\varvec{u}}},\mathbf {p})-(\mathbf {{\varvec{u}}}_{h},\mathbf {p}_{h})\) and using (3.10) we arrive at
Then, noting that obviously
and applying the supremum in (3.12), we find that
Next, employing the second equation of (2.2) and the definition of \(\mathbf {b}\), we deduce that
which yields
Finally, from (3.10) and the first equation of (2.9), we obtain (3.11), and the proof concludes. \(\square \)
The next step consists in deriving suitable upper bounds for the residual term \(\Vert \varvec{u}_h^\mathrm {B}\cdot \varvec{n}-\varvec{u}_h^\mathrm {D}\cdot \varvec{n}\Vert _{-1/2,\Sigma }\) and for \(\Vert \mathbf {E}\Vert _{\mathbf {H}'}\). We begin with the following result.
Lemma 7
There exists \(C_{4}>0\), independent of the meshsizes, such that
Proof
Taking \(\xi _{h}\in \mathrm {Q}_{h}^{\Sigma }\) and then \(\mathbf {p}_{h}\,=\,(0,0,\xi _{h})\in \mathbf {Q}_{h,0}\) in the second equation of (2.9), we find that
which says that each component of \(\varvec{u}_h^\mathrm {B}\cdot \varvec{n}-\varvec{u}_h^\mathrm {D}\cdot \varvec{n}\) belongs to \(\mathrm {Q}_{h}^{\Sigma ,\bot }\) (cf. 3.3). In this way, (3.13) follows from a direct component-wise application of (3.4) (cf. Lemma 6).
We now aim to estimate \(\Vert \mathbf {E}\Vert _{\mathbf {H}'}\). To this end, we first rewrite the functional as follows
where \(\mathbf {E}_{1}\in {\mathbf {H}}_{\mathrm B}({\mathrm {div}};{\Omega _{\mathrm {B}}})', \mathbf {E}_{2}\in \mathbf {H}_0({\mathbf {curl}};{\Omega _{\mathrm {B}}})'\) and \(\mathbf {E}_{3}\in \mathbf {H}_{\mathrm D}(\mathop {\mathrm {div}}\nolimits ;{\Omega _{\mathrm {D}}})'\) are defined by
Notice, from (3.11), that \(\forall {\mathbf {\varvec{v}}}_h:=(\varvec{v}_h^\mathrm {B}, \varvec{z}_h^\mathrm {B}, \varvec{v}_h^\mathrm {D})\in \mathbf {H}_{h}\), there holds
3.3.2 Upper bound for \(\Vert \mathbf {E}_{1}\Vert _{\mathbf {H}_{\mathrm B}({\mathrm {div}};{\Omega _{\mathrm {B}}})'}\)
Given \(\varvec{v}_\mathrm {B}\in \mathbf {H}_{\mathrm B}({\mathrm {div}};{\Omega _{\mathrm {B}}})\), we consider its Helmholtz decomposition established in Lemma 4. More precisely, we let \(w\in \mathrm {H}^{2}({\Omega _{\mathrm {B}}})\) and \(\varvec{\beta }\in \mathbf {H}^{1}_{{\Gamma _{\mathrm {B}}}}({\Omega _{\mathrm {B}}})\) be such that \(\varvec{v}_\mathrm {B}=\nabla w+{\mathbf {curl}}\,\varvec{\beta }\) in \({\Omega _{\mathrm {B}}}\), and
Then, we define the discrete Helmholtz decomposition associated to \(\varvec{v}_h^\mathrm {B}\) as
where \(\Pi _{h}^{{\mathrm B}}\) and \(\mathbf {I}_{h}^{{\mathrm B}}\) are the Raviart–Thomas and Clément operators, respectively, introduced in Sect. 3.1. Then, using from (3.15) that \(\mathbf {E}_{1}(\varvec{v}_h^\mathrm {B})=0\), we can rewrite
Consequently, in what follows we derive suitable upper bounds for the module of the two expressions on the right hand side of (3.17), which are provided by the following two lemmas.
Lemma 8
There exists \(C>0\), independent of meshsizes, such that for each \(w\in \mathrm {H}^{2}{({\Omega _{\mathrm {B}}})}\) there holds
where
Proof
Using the definition of the functional \(\mathbf {E}_{1}\) (cf. 3.14), the identity (3.2), the fact that \(p^{{\mathrm B}}_{h}|_{F}\in P_{0}(F)\) for each \(F\in \mathcal {F}_{h}(\Sigma )\), and the characterisation of \(\Pi _{h}^{{\mathrm B}}\) given in (3.1), we find that
In turn, the fact that \(\nabla w\in \mathrm {H}^{1}({\Omega _{\mathrm {B}}})\) guarantees that \((\nabla w-\Pi ^{{\mathrm B}}_{h}(\nabla w))\cdot \varvec{n}\in \mathrm {L}^{2}(\Sigma )\), and hence
which, together with (3.19), gives
In this way, employing the Cauchy–Schwarz inequality, and the approximation properties of \(\Pi _{h}^{{\mathrm B}}\) given in Lemma 1, we deduce from the above expression that
which yields (3.18) and completes the proof. \(\square \)
Lemma 9
There exists \(C>0\), independent of meshsizes, such that for each \(\varvec{\beta }\in \mathrm {H}^{1}{({\Omega _{\mathrm {B}}})}\) there holds
where
Proof
Given \(\varvec{\beta }\in \mathbf {H}^{1}({\Omega _{\mathrm {B}}})\), we deduce from (3.14) and the identity \({\mathrm {div}}\{{\mathbf {curl}}(\varvec{\beta }-\mathbf {I}_{h}^{{\mathrm B}}\varvec{\beta })\}=0\), that
In turn, thanks to the identities given in [30, Chapter I, Eq. (2.17) and Theorem 2.11], we find that
which gives
Now, integrating by parts in the first term on the right hand side of the last equation, we obtain
Applying Cauchy–Schwarz inequality, Lemma 2, and the uniform boundedness of the number of tetrahedra of the macro-elements \(\Delta _{{\mathrm B}}(T)\) and \(\Delta _{{\mathrm B}}(F)\), we deduce from (3.21) that
which implies (3.20) and ends the proof. \(\square \)
The following Lemma concludes the upper bound for \(\Vert \mathbf {E}_{1}\Vert _{\mathbf {H}_{{\mathrm B}}({\mathrm {div}};{\Omega _{\mathrm {B}}})'}\).
Lemma 10
Assume that there exists a convex domain \(\Xi _{{\mathrm B}}\) such that \({\Omega _{\mathrm {B}}}\subseteq \Xi _{{\mathrm B}}\) and \({\Gamma _{\mathrm {B}}}\subseteq \partial \Xi _{{\mathrm B}}\). Then, there exists \(C_{1}>0\), independent of meshsizes, such that
where \(\widetilde{\Theta }_{{\mathrm B},T}^{2}:=\widetilde{\Theta }_{1,T}^{2}+\widetilde{\Theta }_{2,T}^{2}\), that is
Proof
It follows from (3.18), (3.20), and the stability of the Helmholtz decomposition (3.16). \(\square \)
3.3.3 Upper bounds for \(\Vert \mathbf {E}_{2}\Vert _{\mathbf {H}_0({\mathbf {curl}};{\Omega _{\mathrm {B}}})'}\) and \(\Vert \mathbf {E}_{3}\Vert _{\mathbf {H}_{\mathrm D}(\mathop {\mathrm {div}}\nolimits ;{\Omega _{\mathrm {D}}})'}\)
We first establish the upper bound for \(\Vert \mathbf {E}_{3}\Vert _{\mathbf {H}_{\mathrm D}(\mathop {\mathrm {div}}\nolimits ;{\Omega _{\mathrm {D}}})'}\), which is basically a “mirror reflection” through \(\Sigma \) of Lemma 10.
Lemma 11
Assume that there exists a convex domain \(\Xi _{{\mathrm D}}\) such that \({\Omega _{\mathrm {D}}}\subseteq \Xi _{{\mathrm D}}\) and \({\Gamma _{\mathrm {D}}}\subseteq \partial \Xi _{{\mathrm D}}\). Then, there exists \(C_{3}>0\), independent of the meshsizes, such that
where
Proof
It proceeds exactly as in the proofs of Lemmas 8, 9, and 10, by replacing \({\Omega _{\mathrm {B}}}\), \({\Gamma _{\mathrm {B}}}\), and \(\mathbf {H}_{\mathrm B}({\mathrm {div}};{\Omega _{\mathrm {B}}})\) by \({\Omega _{\mathrm {D}}}\), \({\Gamma _{\mathrm {D}}}\), and \(\mathbf {H}_{\mathrm D}({\mathrm {div}};{\Omega _{\mathrm {D}}})\), respectively. We omit further details. \(\square \)
The upper bound for \(\Vert \mathbf {E}_{2}\Vert _{\mathbf {H}_0({\mathbf {curl}};{\Omega _{\mathrm {B}}})'}\) is provided next. Indeed, the derivation of this bound hinges on the Helmholtz decomposition given in Lemma 5, integration by parts, and the approximation properties of the Clément operators \(I^B_h\) and \(\mathcal N_h\) established in Lemmas 2 and 3, respectively.
Lemma 12
There exists \(C_{2} > 0\), independent of the meshsizes, such that
where
Proof
Given \(\varvec{z}_\mathrm {B}\in \mathbf {H}_0({\mathbf {curl}};{\Omega _{\mathrm {B}}})\), we know from Lemma 5 that there exist \(\varvec{\varphi }\in \mathbf {H}^{1}_{0}({\Omega _{\mathrm {B}}})\) and \(\chi \in \mathrm {H}^{1}_{0}({\Omega _{\mathrm {B}}})\), such that
and
Next, employing the operators \(\mathrm {I}_{h}^{{\mathrm B}}\) and \(\varvec{\mathcal {N}}_{h}\) defined in Sect. 3.1, we introduce the following discrete Helmholtz decomposition
which clearly belongs to \(\mathbf {H}_{h,0}^{{\mathrm B}}\). In this way, and recalling from (3.15) that \(\mathbf {E}_{2}(\varvec{z}_h^\mathrm {B})=0\), it follows that
from which, according to the definition of \(\mathbf {E}_{2}\) (cf. 3.14), we find that
Then, integrating by parts on each T, and noting that \({\mathrm {div}}\,\varvec{\omega }_h^\mathrm {B}\) is zero on T (cf. 2.6, 2.7), we have
In this way, applying Cauchy–Schwarz inequality, the approximation properties of \(\mathrm {I}_{h}^{{\mathrm B}}\) and \(\varvec{\mathcal {N}}_{h}\) given in Lemmas 2 and 3, respectively, the fact that the number of tetrahedra of the macro-elements \(\Delta _{{\mathrm B}}(T)\) and \(\Delta _{{\mathrm B}}(F)\) is uniformly bounded, and the stability estimate (3.23), we get (3.22) and finish the proof. \(\square \)
We end this section by concluding that the reliability of \(\Theta \), that is the upper bound in (3.8), is a straightforward consequence of Theorem 2 and Lemmas 7, 10–12.
3.4 Efficiency
We now devote our attention to the derivation of upper bounds depending on the actual errors associated to the local indicators on each subdomain. For clarity of the analysis we will restrict ourselves to piecewise polynomial forcing terms \(\varvec{f}_{\mathrm {B}}\) and \(\varvec{f}_{\mathrm {D}}\), but we remark that if they are otherwise sufficiently smooth, the error committed from suitable polynomial approximation would produce additional higher order terms in (3.8), explaining the eventual appearance of \(\mathrm{h.o.t}\) in that inequality.
First, and thanks to the incompressibility condition in \({\Omega _{\mathrm {B}}}\) (respectively \({\Omega _{\mathrm {D}}}\)), one has that
The remaining terms in \(\Theta _{{\mathrm B},T}^{2}\) and \(\Theta _{{\mathrm D},T}^{2}\) can be treated very much in the same way as done in [24, 27, 28], where the analysis is based on inverse inequalities found in [19], together with the localisation technique based on tetrahedron-bubble and facet-bubble functions [34]. Such a theory requires further notation and preliminary results collected in what follows.
Given \(T\in \mathcal {T}_{h}\) and \(F\in \mathcal {F}(T)\), let \(\psi _{T}\) and \(\psi _{F}\) denote tetrahedron-bubble and face-bubble functions, respectively (see [33, Eqs. (1.4) and (1.6)]), which satisfy:
-
(i)
\(\psi _{T}\in P_{4}(T),\,\mathrm {supp}(\psi _{T})\subseteq T,\,\psi _{T}=0\) on \(\partial T\), and \(0\le \psi _{T}\le 1\) in T.
-
(ii)
\(\psi _{F}|_{T}\in P_{3}(T),\,\mathrm {supp}(\psi _{F})\subseteq \omega _{F}:=\cup \{T'\in \mathcal {T}_{h}: F\in \mathcal {F}(T')\},\,\psi _{F}=0\) on \(\partial T\backslash \{F\}\), and \(0\le \psi _{F}\le 1\) in \(\omega _{F}\).
In addition, there exists an extension operator \(L:C(F)\rightarrow C(T)\) that satisfies \(L(p)\in P_{k}(T)\) and \(L(p)|_{F}=p\quad \forall p\in P_{k}(F)\), for a given \(k\ge 0\) (see [32]). The vectorial counterpart of L will be denoted \(\mathbf {L}\). Moreover, the following properties hold (where a proof can be found in [32, Lemma 4.1]).
Lemma 13
Given \(k\in \mathbb {N}\cup \{0\}\), there exist \(c_{1}, c_{2},c_{3}>0\), depending only on k and the shape regularity of the triangulations, such that for each \(T\in \mathcal {T}_{h}\) and \(F\in \mathcal {F}(T)\), there hold
The following inverse estimate is also required (see a proof in [19, Theorem 3.2.6]).
Lemma 14
Let \(l,m\in \mathbb {N}\cup \{0\}\) such that \(l\le m\). Then, there exists \(c_4>0\), depending only on k, l, m and the shape regularity of the triangulations, such that for each \(T\in \mathcal {T}_{h}\) there holds
Finally we give two technical lemmas before tackling the derivation of the required upper bounds.
Lemma 15
Let \(\varvec{\zeta }_{h}\in \mathbf {L}^{2}(\Omega )\) be an element-wise polynomial of degree \(k\ge 0\), and let \(\varvec{\zeta }\in \mathbf {L}^{2}(\Omega )\) be such that \({\mathbf {curl}}(\varvec{\zeta })=\mathbf{0}\) in \(\Omega \). Then, there exist \(c_{5},c_{6}>0\), independent of the meshsize, such that
where the set \(\omega _{F}\) is given by \(\omega _{F}:=\cup \{T'\in \mathcal {T}_{h}:\,F\in \mathcal {F}(T')\}\).
Proof
See [24, Lemmas 4.9 and 4.10, respectively]. \(\square \)
Lemma 16
Let \(\varvec{\zeta }_{h}\in \mathbf {L}^{2}(\Omega )\) be an element-wise polynomial of degree \(k\ge 0\), and let \(\varvec{\zeta }\in \mathbf {L}^{2}(\Omega )\) be such that \({\mathrm {div}}(\varvec{\zeta })= 0\) in \(\Omega \). Then, there exist \(c_{7},c_{8}>0\), independent of the meshsize, such that
Proof
Indeed, applying the first inequality given in (3.25), using that \({\mathrm {div}}(\varvec{\zeta })=0\) in \(\Omega \), integrating by parts, and then employing the Cauchy–Schwarz inequality, we get
Now, using the inverse inequality (3.26), and the fact that \(0\le \psi _{T}\le 1\) in T, we find that
which together with (3.31) gives (3.29). The proof of (3.30) corresponds to a slight adaptation of the proof of [7, Lemma 4.6], which makes use of (3.29). \(\square \)
After these preliminary results, we are ready to give local efficiency estimates for several terms associated to the interface.
Lemma 17
There exist constants \(c_{i}>0\), \(i\in \{9,10,11\}\), independent of the meshsizes, such that
-
(a)
\(h_{F}\Vert p_{h}^{{\mathrm B}}-\lambda _h\Vert _{0,F}^{2}\le c_{9}\Big \{\Vert p_{{\mathrm B}}-p_{h}^{{\mathrm B}}\Vert _{0,T_{F}}^{2}+h_{T}^{2}\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T_{F}}^{2}+h_{F}\Vert \lambda -\lambda _h\Vert _{0,F}^{2}\Big \}\), for all \(F\in \mathcal {F}_{h}(\Sigma )\), where \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm B}}\) having F as a face,
-
(b)
\(h_{F}\Vert p_{h}^{{\mathrm D}}-\lambda _h\Vert _{0,F}^{2}\le c_{10}\Big \{\Vert p_{{\mathrm D}}-p_{h}^{{\mathrm D}}\Vert _{0,T_{F}}^{2}+h_{T}^{2}\Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T_{F}}^{2}+h_{F}\Vert \lambda -\lambda _h\Vert _{0,F}^{2}\Big \}\), for all \(F\in \mathcal {F}_{h}(\Sigma )\), where \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm D}}\) having F as a face,
-
(c)
\(h_{F}\Vert \varvec{u}_\mathrm {B}\cdot \varvec{n}-\varvec{u}_\mathrm {D}\cdot \varvec{n}\Vert _{0,F}^{2} \le c_{11}\Big \{\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T_{F}}^{2}+h_{T}^{2}\Vert {\mathrm {div}}\,(\varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B})\Vert _{0,T_{F}}^{2}+\Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T_{F}}^{2}+h_{T}^{2}\Vert {\mathrm {div}}\,(\varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D})\Vert _{0,T_{F}}^{2}\Big \}\), for all \(F\in \mathcal {F}_{h}(\Sigma )\), where \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm B}}\cup \mathcal {T}_{h}^{{\mathrm D}}\) having F as a face.
Proof
Estimates (a) and (b) can be obtained by adapting the proof of [6, Lemma 4.12], whereas (c) follows after a slight modification of the proof in [27, Lemma 3.17] (see also [6, Lemma 4.7]). \(\square \)
Lemma 18
There exist constants \(c_{i}>0\), \(i\in \{12,13\}\), independent of the meshsizes, such that
-
(a)
$$\begin{aligned}&\displaystyle \sum _{F\in \mathcal {F}_{h}(\Sigma )}h_{F}\Vert (\varvec{f}_{\mathrm {B}}-\kappa ^{-1}_{\mathrm {B}}\,\varvec{u}_h^\mathrm {B}-\nu \,{\mathbf {curl}}\,\varvec{\omega }_h^\mathrm {B})\times \varvec{n}-{\mathbf {curl}}_{s}(\lambda _{h})\Vert _{0,F}^{2} \\&\quad \displaystyle \le \, c_{12}\,\left\{ \displaystyle \sum _{F\in \mathcal {F}_{h}(\Sigma )}\Big (\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T_{F}}^{2} +\Vert {\mathbf {curl}}(\varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B})\Vert _{0,T_{F}}^{2}\Big )\,\right. \\&\left. \qquad +\,\Vert \lambda -\lambda _h\Vert _{1/2,\Sigma }^{2}\right\} ,\\ \end{aligned}$$
-
(b)
$$\begin{aligned}&\displaystyle \sum _{F\in \mathcal {F}_{h}(\Sigma )}h_{F}\Vert (\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_h^\mathrm {D})\times \varvec{n}-{\mathbf {curl}}_{s}(\lambda _{h})\Vert _{0,F}^{2}\\&\quad \displaystyle \le \, c_{13}\,\left\{ \displaystyle \sum _{F\in \mathcal {F}_{h}(\Sigma )}\Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T_{F}}^{2}+\Vert \lambda -\lambda _h\Vert _{1/2,\Sigma }^{2}\right\} . \end{aligned}$$
where, given \(F\in \mathcal {F}_{h}(\Sigma )\), \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm B}}\) (respectively \(\mathcal {T}_{h}^{{\mathrm D}}\)) having F as a face.
Proof
The proofs of (a) and (b) follow after a straightforward adaptation of that of [25, Lemma 20], and recalling from [13, Lemma 3.6] that the operator \({\mathbf {curl}}_{s}\) is bounded. \(\square \)
We remark that estimates (a) and (b) provided by the previous lemma are the only nonlocal bounds of the efficiency analysis. However, under an additional regularity assumption on \(\lambda \) we are able to prove the following local bounds.
Lemma 19
Assume that \(\lambda |_{F}\in \mathrm {H}^{1}(F)\), for each \(F\in \mathcal {F}_{h}(\Sigma )\). Then there exist \(c_{14}, c_{15}>0\), independent of the meshsizes, such that for each \(F\in \mathcal {F}_{h}(\Sigma )\) there hold
and
where \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm B}}\) (respectively \(\mathcal {T}_{h}^{{\mathrm D}}\)) having F as a face.
Proof
The derivation of these estimates follows as in the proof of [25, Lemma 21].
\(\square \)
The following three lemmas provide the corresponding upper bounds for the remaining terms defining \(\Theta _{{\mathrm B},T}^{2}\) (cf. 3.5) and \(\Theta _{{\mathrm D},T}^{2}\) ( cf. 3.6).
Lemma 20
There exist positive constants \(c_{i}\), \(i\in \{16,17,18,19\}\), independent of the meshsizes, such that
-
(a)
\(h_{T}^{2}\Vert {\mathbf {curl}}\{\varvec{f}_{\mathrm {B}}-\kappa ^{-1}_{\mathrm {B}}\,\varvec{u}_h^\mathrm {B}-\nu \,{\mathbf {curl}}\varvec{\omega }_h^\mathrm {B}\}\Vert _{0,T}^{2}\le c_{16} \Big \{\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T}^{2}+\Vert {\mathbf {curl}}(\varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B})\Vert _{0,T}^{2}\Big \}\), for all \(T\in \mathcal {T}_{h}^{{\mathrm B}}\).
-
(b)
\(\Vert \varvec{f}_{\mathrm {B}}-\kappa ^{-1}_{\mathrm {B}}\,\varvec{u}_h^\mathrm {B}-\nu \,{\mathbf {curl}}\varvec{\omega }_h^\mathrm {B}\Vert _{0,T}^{2}\le c_{17} \Big \{\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T}^{2}+\Vert {\mathbf {curl}}(\varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B})\Vert _{0,T}^{2}+ h_{T}^{-2} \Vert p_{{\mathrm B}}-p_{h}^{{\mathrm B}}\Vert _{0,T}^{2}\Big \}\), for all \(T\in \mathcal {T}_{h}^{{\mathrm B}}\).
-
(c)
\(\displaystyle h_{F}\Vert \llbracket (\varvec{f}_{\mathrm {B}}-\kappa ^{-1}_{\mathrm {B}}\,\varvec{u}_h^\mathrm {B}-\nu \,{\mathbf {curl}}\varvec{\omega }_h^\mathrm {B})\times \varvec{n}\rrbracket \Vert _{0,F}^{2}\le c_{18} \Big \{\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,\omega _{F}}^{2}+\Vert {\mathbf {curl}}(\varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B})\Vert _{0,\omega _{F}}^{2}\Big \}\), for all \(F\in \mathcal {F}_{h}({\Omega _{\mathrm {B}}})\), where \(\omega _{F}:=\cup \{T'\in \mathcal {T}_{h}^{{\mathrm B}}:\,F\in \mathcal {F}(T')\}\).
-
(d)
\(h_{F}\Vert (\varvec{f}_{\mathrm {B}}-\kappa ^{-1}_{\mathrm {B}}\,\varvec{u}_h^\mathrm {B}-\nu \,{\mathbf {curl}}\,\varvec{\omega }_h^\mathrm {B})\times \varvec{n}\Vert _{0,F}^{2}\le c_{19}\,\Big \{\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T_{F}}^{2}+\Vert {\mathbf {curl}}(\varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B})\Vert _{0,T_{F}}^{2}+\mathrm{h.o.t}\Big \}\), for all \(F\in \mathcal {F}_{h}({\Gamma _{\mathrm {B}}})\), where \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm B}}\) having F as a face, and
$$\begin{aligned} \mathrm{h.o.t}:=h_{T_{F}}^{4}\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T_{F}}^{2}+h_{T_{F}}^{4}\Vert {\mathbf {curl}}(\varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B})\Vert _{0,T_{F}}^{2}+h_{T_{F}}^{2}\Vert p_{{\mathrm B}}-p_{h}^{{\mathrm B}}\Vert _{0,T_{F}}^{2}. \end{aligned}$$
Proof
Since \({\mathbf {curl}}(\varvec{f}_{\mathrm {B}}-\kappa ^{-1}_{\mathrm {B}}\,\varvec{u}_\mathrm {B}-\nu \,{\mathbf {curl}}\varvec{\omega }_\mathrm {B})={\mathbf {curl}}(\nabla p_{{\mathrm B}})=\mathbf{0}\) in \({\Omega _{\mathrm {B}}}\), to derive (a) and (c) it suffices to apply the estimates (3.27) and (3.28) (cf. Lemma 15), respectively, to
On the other hand, reasoning similarly as in the proof of [29, Lemma 5.14], we get
from which, it is easy to deduce the estimate (b). For the proof of (d), we set \(\varvec{\zeta }\) and \(\varvec{\zeta }_{h}\) as in (3.32). Given \(F\in \mathcal {F}_{h}(\Gamma )\) we denote \(\varvec{\chi }_{F}:=\varvec{\zeta }_{h}\times \varvec{n}\) on F. Then, applying the second inequality given in (3.25), and the extension operator \(\mathbf {L}:\mathbf {C}(F)\rightarrow \mathbf {C}(T)\), we find that
Now, integrating by parts, it follows that
Next, applying the Cauchy–Schwarz inequality, the inverse estimate (3.26), and the preliminary bound for \(\Vert {\mathbf {curl}}(\varvec{\zeta }_{h})\Vert _{0,T_{F}}\) (cf. 3.27), we deduce that
In turn, recalling that \(0\le \psi _{F} \le 1\) in F, and employing the third inequality in (3.25), we can write
Finally, using (3.35), the definitions of \(\varvec{\zeta }\) and \(\varvec{\zeta }_{h}\) (cf. 3.32), the preliminary estimate (3.33), and the fact that \(h_{F}\le h_{T_{F}}\), we deduce from (3.34) that
where
which gives (d), and ends the proof. \(\square \)
Lemma 21
There exist \(c_{i}>0\), \(i\in \{20,21,22,23\}\), independent of the meshsizes, such that
-
(a)
\(h_{T}^{2}\Vert {\mathbf {curl}}\{\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_h^\mathrm {D}\}\Vert _{0,T}^{2}\le c_{20}\, \Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T}^{2}\qquad \forall \,\,T\in \mathcal {T}_{h}^{{\mathrm D}}\),
-
(b)
\(h_{T}^{2}\Vert \varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_h^\mathrm {D}\Vert _{0,T}^{2}\le c_{21} \,\Big \{h_{T}^{2}\,\Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T}^{2}+\,\Vert p_{{\mathrm D}}-p_{h}^{{\mathrm D}}\Vert _{0,T}^{2}\Big \}\qquad \forall \,\,T\in \mathcal {T}_{h}^{{\mathrm D}}\),
-
(c)
\(\displaystyle h_{F}\Vert \llbracket (\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_h^\mathrm {D})\times \varvec{n}\rrbracket \Vert _{0,F}^{2}\le c_{22} \Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,\omega _{F}}^{2}\) for all \(F\in \mathcal {F}_{h}({\Omega _{\mathrm {D}}})\), where the set \(\omega _{F}\) is given by \(\omega _{F}:=\cup \{T'\in \mathcal {T}_{h}^{{\mathrm D}}:\,F\in \mathcal {F}(T')\}\),
-
(d)
\(h_{F}\Vert (\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_h^\mathrm {D})\times \varvec{n}\Vert _{0,F}^{2}\le c_{23}\,\Big \{\Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T_{F}}^{2}+\Vert p_{{\mathrm D}}-p_{h}^{{\mathrm D}}\Vert _{0,T_{F}}^{2}+\mathrm{h.o.t}\Big \}\) for all \(F\in \mathcal {F}_{h}({\Gamma _{\mathrm {D}}})\), where \(T_{F}\) is the tetrahedron of \(\mathcal {T}_{h}^{{\mathrm D}}\) having F as a face, and
$$\begin{aligned} \mathrm{h.o.t}:=h_{T_{F}}^{4}\Vert \varvec{u}_\mathrm {D}-\varvec{u}_h^\mathrm {D}\Vert _{0,T_{F}}^{2}+h_{T_{F}}^{2}\Vert p_{{\mathrm D}}-p_{h}^{{\mathrm D}}\Vert _{0,T_{F}}^{2}. \end{aligned}$$
Proof
Thanks to the fact that \({\mathbf {curl}}(\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_\mathrm {D})={\mathbf {curl}}(\nabla p_{{\mathrm D}})=\mathbf{0}\) in \({\Omega _{\mathrm {D}}}\), (a) and (c) can be obtained applying (3.27) and (3.28), respectively, to \(\varvec{\zeta }:=\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_\mathrm {D}\) and \(\varvec{\zeta }_{h}:=\varvec{f}_{\mathrm {D}}-\kappa ^{-1}_{\mathrm {D}}\,\varvec{u}_h^\mathrm {D}\). The remaining estimates follow analogously to the proofs of (b) and (d) in Lemma 20. \(\square \)
We next turn to the derivation of local efficiency estimates for the residual expressions defining \(\widehat{\Theta }_{{\mathrm B},T}^{2}\).
Lemma 22
There exist \(c_{i}>0, i\in \{24,25,26\}\), independent of the meshsizes, such that
-
(a)
\(h_{T}^{2}\,\Vert \varvec{\omega }_h^\mathrm {B}-{\mathbf {curl}}\varvec{u}_h^\mathrm {B}\Vert _{0,T}^{2}\le c_{24}\,\Big \{ \Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T}+h_{T}^{2}\,\Vert \varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B}\Vert _{0,T}\Big \}\),
-
(b)
\(\displaystyle h_{F}\,\Vert \llbracket \varvec{u}_h^\mathrm {B}\times \varvec{n}\rrbracket \Vert _{0,F}^{2}\le c_{25}\,\sum _{T\subseteq \omega _{F}}\Big \{\Vert \varvec{u}_\mathrm {B}-\varvec{u}_h^\mathrm {B}\Vert _{0,T}^{2}+h_{T}^{2}\Vert \varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B}\Vert _{0,T}^{2}\Big \}\,\forall \,F\in \mathcal {F}_{h}({\Omega _{\mathrm {B}}})\),
-
(c)
\(h_{F}\,\Vert \llbracket \varvec{\omega }_h^\mathrm {B}\cdot \varvec{n}\rrbracket \Vert _{0,F}^{2}\le c_{26}\,\Vert \varvec{\omega }_\mathrm {B}-\varvec{\omega }_h^\mathrm {B}\Vert _{0,\omega _{F}}^{2}\).
Proof
Regarding (a), let us denote \(\varvec{\chi }_{T}:=\varvec{\omega }_h^\mathrm {B}-{\mathbf {curl}}\,\varvec{u}_h^\mathrm {B}\) in a generic \(T\in \mathcal {T}_h\). Applying the first estimate of (3.25) to \(\varvec{\chi }_{T}\), and then using that \({\mathbf {curl}}\varvec{u}_\mathrm {B}=\varvec{\omega }_\mathrm {B}\) in \({\Omega _{\mathrm {B}}}\), we find that
Next, integrating by parts in the first term on the right hand side of the last identity, and recalling that \(\psi _{T}\) vanishes on \(\partial T\), we obtain
Then, applying Cauchy–Schwarz inequality and (3.26), we deduce from (3.36) that
In this way, using that \(0\le \psi _{T}\le 1\) in T, we get
which gives (a). Estimate (b) can be derived by adapting the arguments in the proof of [2, Lemma 4.15]. Finally, since \({\mathrm {div}}(\varvec{\omega }_\mathrm {B})={\mathrm {div}}({\mathbf {curl}}\,\varvec{u}_\mathrm {B})=0\) in \({\Omega _{\mathrm {B}}}\), for the derivation of (c), it suffices to apply (3.30) to \(\varvec{\zeta }:=\varvec{\omega }_\mathrm {B}\) and \(\varvec{\zeta }_{h}:=\varvec{\omega }_h^\mathrm {B}\). \(\square \)
We end this section by observing that the term \(h_{F}\Vert \lambda -\lambda _h\Vert _{0,F}^{2}\) appearing in Lemma 17 (items (a) and (b)), is bounded as follows:
Therefore the efficiency of \(\Theta \) is a direct consequence of (3.24) and Lemmas 17–21.
4 Numerical results
In this section we provide two computational tests aimed at illustrating the properties of the estimator \(\Theta \) introduced in Sect. 3.2. All linear systems are solved with the distributed multifrontal direct solver MUMPS.
Example 1
For our first test we design a mesh convergence example using two sets of closed-form solutions, and performing uniform and adaptive mesh refinement. The Darcy and Brinkman sub-domains consist of two boxes \({\Omega _{\mathrm {D}}}=(-0.5,0.5)^3\), \({\Omega _{\mathrm {B}}}= (-0.125,0.125)^2 \times (-0.4,0.4)\). The model parameters are \(\kappa _{\mathrm {B}}^{-1}= 10\), \(\kappa _{\mathrm {D}}^{-1}=50\), \(\nu =0.01\). The convergence of the method is assessed by computing errors between the following manufactured smooth exact solutions
and their finite element approximations using a \(\mathbb {RT}_0-\mathbb {ND}_1-\mathbb {RT}_0-\mathbf {P}_0-\mathbf {P}_0-\mathbf {P}_1\) family.
The domains are discretised into a series of nested uniform triangulations, where errors, experimental convergence rates, and effectivity indexes will be defined as
with e and \(\hat{e}\) denoting errors associated to two consecutive meshes of sizes h and \(\hat{h}\), and being associated to methods having N and \(\hat{N}\) degrees of freedom, respectively. The first two parts of Table 1 show optimal convergence for all fields under either adaptive or uniform mesh refinement.
Secondly, we regard the same domains but manufacture an exact pressure that is singular near one wall of \({\Omega _{\mathrm {D}}}\) (the singularity being located at \((x_a,x_b,x_c)=(0,0,-0.55)\)):
We expect that the convergence is hindered by the lower regularity of the exact solution. This is indeed evidenced in the third block of Table 1, where we see an oscillating effectivity index and a very low convergence, especially so for the Darcy pressure and the Lagrange multiplier. The optimal character of the error decay is however restored when we use an adaptive mesh refinement strategy (see the last section of the table). We also confirm that the error indicator \(\Theta \) performs well even if the fluid viscosity \(\nu \) has a considerable variation (see Table 2). Intermediate adapted meshes and some components of the approximate solution are displayed in Fig. 1.
Example 2
Next we turn to the simulation of the flow behaviour within a composite domain \(\Omega = (0,2)\times (0,0.2)\times (0.75)\). A smooth interface exists between the Darcy and Brinkman subdomains, where the Brinkman part is on top (see related test cases in [1, 4, 10, 14]). For this problem we assume a uniform current flow on the \(x_1-\)direction and the presence of gravity, so \(\varvec{f}_{\mathrm {B}}=\varvec{f}_{\mathrm {D}}= (0.25,0,-0.1)^T\). In addition, we take a dimensional parameters specified as \(\kappa _{\mathrm {B}}^{-1}= 1\), \(\nu =0.01\), and \(\kappa _{\mathrm {D}}^{-1}=8 + 800\eta (x_1,x_2,x_3)\), where \(\eta \) is the sum of characteristic functions on 20 balls of radius 1E-4, located randomly in \({\Omega _{\mathrm {D}}}\) and representing obstacles of much lower permeability.
The boundary conditions are set as follows: on the face \(x_1=0\) we impose a unitary normal inflow velocity \(\varvec{u}_\mathrm {B}\cdot \varvec{n}= 1\), on the bottom and top faces we set slip velocity conditions \(\varvec{u}_\mathrm {B}\cdot \varvec{n}= 0\) and \(\varvec{u}_\mathrm {D}\cdot \varvec{n}=0\) (for the Brinkman and Darcy boundaries, respectively), and on the remaining parts of the boundary we do not force velocity nor pressure. On the interface we impose zero tangential vorticity and the transmission conditions analyzed in the paper. We now use the method based on the \(\mathbb {RT}_1-\mathbb {ND}_2-\mathbb {RT}_1-\mathbf {P}_1-\mathbf {P}_1-\mathbf {P}_2\) family, and a penalisation approach is used to impose zero-mean value of the Brinkman pressure. In Fig. 2 we present the sketch of the domains and interface, the obtained approximate solutions and snapshots of two adaptive meshes produced following the a posteriori error estimator. All fields are well-resolved, even with coarse grids.
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Acknowledgements
This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and project Anillo ACT1118 (ANANUM); by the Ministery of Education through the project REDOC.CTA of the Graduate School, Universidad de Concepción; by Centro de Investigación en Ingeniería Matemática (CI\(^2\)MA), Universidad de Concepción; by Vicerrectoría de Investigación, project 540-B7-233, Sede de Occidente, Universidad de Costa Rica; and by the EPSRC through the research grant EP/R00207X/1.
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Álvarez, M., Gatica, G.N. & Ruiz-Baier, R. A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem. Calcolo 54, 1491–1519 (2017). https://doi.org/10.1007/s10092-017-0238-z
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DOI: https://doi.org/10.1007/s10092-017-0238-z
Keywords
- Brinkman–Darcy equations
- Vorticity-based formulation
- Mixed finite element methods
- A posteriori error analysis