## Abstract

In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of
*n*-dimensional fluids,

## 1 Introduction

Free convective flows can be found in a wide amount of settings throughout nature and industry, for instance, in mantle convection, stratified oceanic flows and the cooling of electronic devices, to name a few. Many of these processes can be modeled by coupling the equations of continuity, momentum (Navier–Stokes) and energy using the Boussinesq approximation, which (in this context) assumes the density of the fluid to be constant in all terms of the equations, except in the buoyancy term of the momentum equation, where a linear dependence is considered. Nevertheless, other properties may also vary with temperature, as is the case of, for example, viscosity and thermal conductivity in oils and nanofluids, which poses a significant effect on the fluid flow. In this regard, several finite element methods to approximate the solution to this and related problems, both with constant and temperature-dependent properties have been proposed (see [2, 3, 4, 5, 6, 8, 12, 13, 14, 22, 23, 25, 26, 27] and the references therein).

In particular, the authors in [22, 23] propose finite element methods based on primal formulations of the Boussinesq system. The first one deals with the problem in its primitive variables, while the second one introduces the normal heat flux through the boundary as an additional variable to achieve conformity of the scheme. Nonetheless, both methods are proved to be optimally convergent whenever the exact solution is smooth enough, and the data and the

The purpose of this work is to extend the analysis and results of [4] by deriving now
an augmented fully-mixed finite element method for the Boussinesq problem, but considering this time both the viscosity and the thermal conductivity of the fluid as temperature-dependent functions, and mixed thermal boundary conditions. To this end, we consider again the mixed formulation of the Navier–Stokes equations in [4], to then, using this same approach, construct a mixed formulation for the energy equation. More precisely, we consider the temperature gradient and pseudoheat vector as additional variables, which together with the temperature, rate of strain, pseudostress, velocity and vorticity comprise the unknowns of the problem. At this point, we remark that the main difference with respect to [13] is the consideration in this work of temperature-dependent parameters, which also becomes the cause of defining the rate of strain and temperature gradient in addition to the variables defined in the aforementioned work (recall from [2] that the vorticity appears in this formulation because of the consideration of a more physical version of the Cauchy stress tensor). That being said, part of our analysis follows basically the same uncoupling and fixed-point strategies from [2, 4, 5, 12, 13], reason why we do not provide all details but
make the proper references when it corresponds. At a continuous level, we prove that the uncoupled problems are well posed thanks to the Lax–Milgram theorem, and then we prove that the fixed-point operator admits a unique fixed point by means of the Banach fixed-point theorem, whenever the data is sufficiently small and assuming that the exact solution has a slightly higher regularity than the one the well-posedness results provide. Then, following these same steps, we provide a well-posedness result for the Galerkin scheme where one of the key features of this work can be appreciated: there is no need to impose inf-sup conditions on the discrete analysis, which gives us the freedom to choose any combination of finite element subspaces. In particular, we approximate the pseudostress and pseudoheat variables using Raviart–Thomas elements of order *k*, the velocity and temperature using Lagrange elements of order

Finally, we consider worthwhile to mention other features of this fully-mixed method. First, as Dirichlet conditions appear naturally in mixed formulations, it is not necessary to define boundary unknowns to achieve conformity in the scheme (see the Lagrange multiplier defined in [2, 4]), thus unifying the analysis of the uncoupled problems and simplifying the computational implementation of this method. Also, as not only the velocity and temperature of the fluid are part of the solution but also their gradients, many other physical variables of interest can be computed as a simple post-process without requiring numerical differentiations that could deteriorate the good quality of the results. Nevertheless, it is also fair to mention that the use of dual-mixed approaches, and the consequent increase in the number of unknowns, yields much larger linear and nonlinear systems to be solved at the discrete level, and hence, the devising of more efficient numerical methods for computing their solution arises as an unavoidable need. In this regard, we believe that the use of Hybridizable Discontinuous Galerkin Methods (HDG) specially designed for nonlinear problems (see, e.g., [18]), appears as a very attractive possibility to be explored in a separate work.

### 1.1 Outline

The rest of this paper is organized as follows. First, we end this section by introducing some notation that will be used throughout the paper. Then, in Section 2, we set the Boussinesq problem with temperature-dependent viscosity and thermal conductivity functions, and introduce the new variables that will allow us to construct a fully-mixed formulation. Next, in Section 3, we uncouple the problem using a fixed-point argument. The uncoupled problems are then analyzed by means of the Lax–Milgram theorem, and existence of a unique fixed point is proved by fulfilling the hypotheses of the Banach fixed-point theorem. Later, in Section 4 these techniques are used to prove the well-posedness of the corresponding Galerkin scheme, but this time using the Brouwer fixed-point theorem, and then we make a specific choice of finite element subspaces. Finally, in Section 5 we derive some a priori error estimates by suitably applying the usual Strang’s lemmas for linear problems, to then in Section 6 illustrate the good performance of this augmented fully-mixed finite element method and confirm the theoretical rates of convergence through several numerical examples in two and three dimensions.

### 1.2 Preliminaries

Let us denote by

In addition, for any tensor fields

where

equipped with the usual norm

is a standard Hilbert space in the realm of mixed problems. Finally, in what follows, *C*, with or without subscripts, bars, tildes or hats, to mean generic positive constants independent of the discretization parameters, which may take different values at different places.

## 2 The Model Problem

In this section, we first introduce the Boussinesq problem with its original unknowns, to then define suitable new variables that will later allow us to construct a fully-mixed formulation.

### 2.1 The Original Formulation

Let us consider the flow of a non-isothermal, incompressible, Newtonian fluid with varying viscosity and thermal conductivity within a region Ω. Then, under the Boussinesq approximation, the problem reads: Find a velocity *p* and a temperature φ such that

(2.1)

where *k* are Lipschitz continuous functions, that is, there exist

### 2.2 Introduction of New Variables

Let us first consider the spaces

and, in a similar way to [4], define the following variables, known respectively as the rate of strain, pseudostress and vorticity tensors:

where

(2.2)

Notice that the continuity equation is implicitly present in (2.2b), and it suggests us that rate of strain tensor

Next, in order to construct a mixed formulation for the energy equation, we follow the approach taken in [9, 13] and define

which from now on will be called “pseudoheat”. In addition, analogously to the mixed formulation for the momentum equation, we consider the temperature gradient as another new variable

Therefore, the energy equation (2.1c) can be rewritten in these terms as

(2.3)

where the Neumann condition (2.1f) has been converted to (2.3e) thanks to the no-slip condition

In this way, the Boussinesq problem (2.1) can now be seen as the set of equations (2.2) and (2.3). Then a mixed formulation of each one of them can be constructed upon integration by parts of (2.2a) and (2.3a) when multiplied by proper test functions, which is the purpose of the next section.

## 3 The Continuous Problem

We now turn to the construction and analysis of a fully-mixed formulation for the Boussinesq problem introduced in the previous section.

### 3.1 An Augmented Fully-Mixed Formulation

First, we recall from [4] the equations corresponding to the mixed formulation of the momentum equation:

where

Then, to obtain a mixed formulation for the energy equation, we multiply equation (2.3a) by a test function

and integrate by parts. Using the boundary condition (2.3d), we get

where

Notice that, due to the second term in both (3.2) and (3.5), we require the velocity and temperature to have (weak) bounded derivatives as shown in the following inequalities, which can be obtained by using the Hölder inequality and the continuous injections

where

for the momentum and continuity equations, and

for the energy equations, where

### Remark 3.1.

We now stress that the incorporation of equations (3.9)–(3.12),
which are extracted directly from [4], allows us to establish the ellipticity
of the resulting bilinear form (cf. proof of Lemma 3.3 below). The same reason applies
to the introduction of (3.13)–(3.16) (cf. proof of Lemma
3.4 below). In turn, the concept “*redundant*” employed here refers to the fact
that the equations yielding these augmented terms have already been considered before for the derivation
of the weak formulation, but tested differently to the way they are tested now in
(3.9)–(3.12) and (3.13)–(3.16).

In this way, denoting by

and

the augmented fully-mixed formulation for this Boussinesq problem is: Find

(3.17)

where, given

for all

for all

for all

### Remark 3.2.

Before we continue, let us have a brief look at what the energy equation (2.1c) would have looked like if we had considered instead a heat-temperature mixed formulation. Indeed, if we define

equation (2.1c) can be rewritten as

Then, multiplying the first equation by a test function

for all

where

In the upcoming sections, we analyze problem (3.17) using fixed-point strategies from [2, 4, 5, 13]. More precisely, in Section 3.2 we rewrite (3.17) as a fixed-point problem, in Section 3.3 we prove the main results stated and employed in Section 3.2, and then in Section 3.4 we establish sufficient conditions for existence and uniqueness of this fixed point.

### 3.2 The Fixed-Point Approach

First, let us define

where

In turn, consider the operator

where

The well-definedness of the mappings

### Lemma 3.3.

*Assume that for *

*where *

### Lemma 3.4.

*Assume that for *

*Then there exists *

Consequently, we can define the operator

and look at (3.17) as the fixed-point problem: Find

Indeed, we begin the corresponding analysis by showing next that the operator *r*, with

### Lemma 3.5.

*Given *

*Assume that the stabilization parameters *

*with *

### Proof.

Let

### 3.3 Proofs of Lemmas 3.3 and 3.4

As usual, we consider

and

### Proof of Lemma 3.3.

Notice that an analogous result has been proved in [4, Lemma 2.3] for the case where a non-homogeneous velocity boundary condition is imposed, however, since we want to find

We begin by analyzing the ellipticity of the bilinear form

Then, using the bounds for the viscosity and the Cauchy–Schwarz and Young inequalities, we obtain for
any

Note here, in particular, that in order to get positive quantities multiplying

whereas the norm of

where

we deduce the existence of a positive constant

The rest of the proof is identical to [4, Lemma 2.3], but we recall it for completion purposes.
Thus, the foregoing inequality, the definition of

and then, we easily see that

provided that

that is,

thus proving ellipticity for

Therefore, by the Lax–Milgram theorem, there exists a unique

Next, we recall the following Poincaré-type inequality that will help us to prove the ellipticity of the
bilinear form defining the operator

### Lemma 3.6.

*There exists *

### Proof.

See [20, Theorem 5.11.2]. ∎

### Proof of Lemma 3.4.

Let

thus obtaining the existence of a positive constant

For

Hence, from the previous two equations, there exists a positive constant denoted by

Next, to prove that

Then, using the bounds for the thermal conductivity function, the Cauchy–Schwarz and Young inequalities, we get for any

Hence, applying Lemma 3.6 and defining the constants

there exists

which, together with (3.21) and inequality (3.8), allows us to write

Therefore, we easily see that

provided that

that is,

thus proving ellipticity for

and

and then, using the continuous injection from

such that

Since

At this point we remark that, while (3.36) and (3.39) guarantee
the ellipticities of

We end this subsection by commenting that for computational purposes, a particular choice of stabilization
parameters has to be made. Hence, we first consider the middle points of the intervals for

Notice that

### 3.4 Solvability Analysis of the Fixed-Point Problem

In this subsection, we pursue to comply with the hypotheses of the Banach fixed-point theorem to ensure existence and uniqueness of a fixed point. More precisely, the main results of this subsection are stated as follows.

### Lemma 3.7.

*Let *

*with *

*where*

*and *

### Theorem 3.8.

*Let *

*r*(cf. (3.31)). Assume that the stabilization parameters

(3.42)

*with constants *

*and*

In order to prove Lemma 3.7 and Theorem 3.8, we first recall from (3.33)
that

Therefore, it only remains to prove that

and (following (3.28))

and on the other hand, that

and (following (3.29))

Here, *r* of its

### Lemma 3.9.

*Let *

*r*, such that

(3.46)

*for all *

### Proof.

See [4, Lemma 2.6]. ∎

We stress here that Lemma 3.9 makes use precisely of the regularity assumption (3.44)
and the fact, that under the specified range of ε,

### Lemma 3.10.

*Let *

*for all *

### Proof.

Let

Then, defining the constants

the result (3.47) holds with

In a similar way, we show next that the operator

### Lemma 3.11.

*Let *

*for all *

### Proof.

Let

it follows that

for any

The last term in the previous inequality can be easily split using (3.8), that is,

whereas for the first term we use the Hölder inequality to show that

with

and hence, there exists

In this way,

and (3.51) now yields

Since

such that

for any

thus obtaining (3.48) with

Consequently, the proof of Lemma 3.7, that is, the Lipschitz-continuity
of

### Proof of Lemma 3.7.

Let

The bound for the first term comes directly from (3.47),
whereas for the second one, we first use the Lipschitz-continuity of

Summarizing, Lemma 3.5 ensures us that

### Proof of Theorem 3.8.

Notice that (3.42a) ensures that both (3.32) and (3.43)
hold, so that

## 4 The Galerkin Scheme

We advocate in this section to present the Galerkin scheme for the continuous problem (3.17), whose well-posedness will be proved using the same steps and methods as in the previous section.

### 4.1 Preliminaries

Let us consider *T* when

and

Hence, according to the continuous formulation (3.17), the Galerkin scheme reads: Find

(4.1)

where the forms

We will see that it is possible to establish sufficient conditions for well-posedness of (4.1)
in the same form they were established for the continuous problem (3.17). To this end,
we now split the discrete formulation into the two corresponding mixed formulations.
In fact, we first set

where

In turn, we let

where

Therefore, by introducing the operator

we can rewrite (4.1) as the fixed-point problem: Find

In this case, existence of a fixed point for this problem will be proved by means of the Brouwer fixed-point theorem, which we recall next.

### Theorem 4.1 (Brouwer).

*Let W be a compact and convex subset of a finite-dimensional Banach space, and let *

### Proof.

See [10, Theorem 9.9-2]. ∎

### 4.2 Solvability Analysis

We first study under which conditions

### Lemma 4.2.

*Assume that for *

*where *

*with *

### Proof.

It follows from a direct application of the Lax–Milgram theorem to (4.2) in the same way it was applied in Lemma 3.3. ∎

### Lemma 4.3.

*Assume that for *

*Then, for each *

*with *

### Proof.

It also follows from an application of the Lax–Milgram theorem to (4.3) in the same way it was applied in Lemma 3.4. ∎

Analogously to the continuous case, the previous two lemmas provide the well-definedness of the operator

### Lemma 4.4.

*Given *

*Assume that the stabilization parameters *

*Moreover, *

We now turn to prove the continuity of

### Lemma 4.5.

*Let *

*r*, such that

*for all *

### Proof.

It comes from [4, Lemma 2.6], but employing an

### Lemma 4.6.

*Let *

*for all *

### Proof.

As in the previous lemma, the proof is based on the one for its continuous counterpart Lemma 3.11, but just taking

Consequently, we have the following result for the operator

### Lemma 4.7.

*Given *

*r*(cf. (4.5)), and assume that (3.41) holds. Then

*for all *

### Proof.

Let

and since (3.41) holds,

where

Having proved that

### Theorem 4.8.

*Let *

*r*(cf. (4.5)). Assume the stabilization parameters

*and*

### 4.3 Specific Finite Element Subspaces

An interesting point to realize in this fully-mixed approach with respect to the finite-dimensional subspaces
is that we have not imposed any kind of inf-sup conditions as in [2, 4],
or any other requirement than being finite-dimensional, which give us the chance to freely choose these subspaces.
In particular, the most natural finite element subspaces of *k* as

where according to the terminology described in Section 1, *k*,
and the velocity and temperature by Lagrange elements of order *k*, that is,

According to [7, 16], their corresponding approximation properties are as follows.

### $(\mathbf{AP}_{h}^{\mathbf{t}})$.

There exists a constant *h*, such that for each

### $(\mathbf{AP}_{h}^{\boldsymbol{\sigma}})$.

There exists a constant *h*, such that for each

### $(\mathbf{AP}_{h}^{\mathbf{u}})$.

There exists a constant *h*, such that for each

### $(\mathbf{AP}_{h}^{\boldsymbol{\gamma}})$.

There exists a constant *h*, such that for each

### $(\mathbf{AP}_{h}^{\boldsymbol{\zeta}})$.

There exists a constant *h*, such that for each

### $(\mathbf{AP}_{h}^{\mathbf{p}})$.

There exists a constant *h*, such that for each

### $(\mathbf{AP}_{h}^{\varphi})$.

There exists a constant *h*, such that for each

## 5 A Priori Error Analysis

Let

(5.1)

and

(5.2)

In addition, we denote as usual

Then the main result of this section reads as follows.

## Theorem 5.1.

*Assume that the data satisfy*

*where *

*h*, such that the following Céa estimate holds:

Similar to [2, Lemma 5.3] and [4, Lemma 4.2], we will apply the
Strang Lemma to the pair of equations (5.1) and (5.2) separately,
to then join the resulting estimates to derive the Céa estimate (5.4).
In this regard, we emphasize that, given arbitrary

## Lemma 5.2 (Strang).

*Let V be a Hilbert space, let *

*In turn, let *

*Then, for each *

*where *

## Proof.

See [11, Theorem 11.1]. ∎

## Lemma 5.3.

*Let *

## Proof.

See [4, Lemma 4.2]. ∎

## Lemma 5.4.

*Let *

(5.5)

## Proof.

It is clear from (3.29) that

where

In what follows we make use of the boundedness constants

which back into (5.6) gives (5.5), concluding this way the proof. ∎

As a result of the previous two lemmas, we have a preliminary estimate for the error:

(5.7)

where

all being positive constants independent of the discretization parameters. Thus, we first bound the terms

inequality (5.7) becomes

thus leading us to the main result of this section, that is, to Theorem 5.1.
In fact, thanks to (5.3), the estimate (5.8)
readily implies (5.4) with

Consequently, when using the finite element subspaces (4.9)–(4.15), the following can be established regarding the rates of convergence of the method.

## Lemma 5.5.

*In addition to the hypotheses of Theorems 3.8, 4.8 and 5.1, assume that there exists *

*h*, such that

## 6 Numerical Results

We now present two examples that will illustrate the performance of the augmented fully-mixed finite element
method (4.1) with the subspaces indicated in (4.9)–(4.15)
on a set of quasiuniform triangulations. This means that, given an integer *k*, and

where tol is a specified tolerance.

Let us first define the error per variable

as well as their corresponding rates of convergence

where *h* and *e* and

### 6.1 Example 1: Two-Dimensional Smooth Exact Solution

In our first example, we consider

Dirichlet boundary conditions will be imposed on

where

with

and

Concerning the stabilization parameters, these are taken as pointed out at the end of Section 3.3,
with

In Figure 1 we display part of the solution obtained with fully-mixed finite element
method using a first-order approximation and 1,409,884 DOF. Notice that not only we are able to recover
the original unknowns but also to compute further variables of physical interest.
In turn, Tables 1 and 2 show the convergence history for a sequence
of quasi-uniform mesh refinements, thus confirming the rates of convergence predicted by
Lemma 5.5, that is, when using first and second-order finite elements,
and considering that the exact solution is smooth enough, whence the method converges with
orders *h* approaches 0, as Lemma 5.5
established. This remark is certainly valid as well for the other tables reported in the
present Section 6.

Finite Element: | ||||||||

DOF | ||||||||

1,816 | 4.5307 | 15.6976 | 7.7257 | 2.3619 | 14.3622 | 0.4298 | 1.5650 | 0.3698 |

6,972 | 2.1933 | 7.9406 | 3.3074 | 1.0874 | 8.8796 | 0.2124 | 0.7864 | 0.1758 |

27,052 | 1.0947 | 3.9421 | 1.6088 | 0.5396 | 6.0695 | 0.1022 | 0.3809 | 0.0831 |

107,142 | 0.5331 | 2.0064 | 0.8011 | 0.2678 | 2.5649 | 0.0517 | 0.1962 | 0.0426 |

431,398 | 0.2581 | 0.9882 | 0.3925 | 0.1216 | 1.5602 | 0.0259 | 0.0970 | 0.0211 |

1,707,922 | 0.1271 | 0.4937 | 0.1947 | 0.0613 | 0.7027 | 0.0127 | 0.0482 | 0.0105 |

h | ||||||||

0.4129 | – | – | – | – | – | – | – | – |

0.1940 | 0.9605 | 0.9023 | 1.1233 | 1.0270 | 0.6366 | 0.9331 | 0.9111 | 0.9845 |

0.0995 | 1.0402 | 1.0482 | 1.0788 | 1.0490 | 0.5695 | 1.0951 | 1.0851 | 1.1221 |

0.0527 | 1.1333 | 1.0638 | 1.0982 | 1.1032 | 1.3567 | 1.0736 | 1.0453 | 1.0521 |

0.0311 | 1.3744 | 1.3419 | 1.3519 | 1.4961 | 0.9419 | 1.3087 | 1.3350 | 1.3342 |

0.0150 | 0.9729 | 0.9535 | 0.9635 | 0.9411 | 1.0960 | 0.9778 | 0.9600 | 0.9503 |

Finite Element: | ||||||||

DOF | ||||||||

5,812 | 0.9568 | 3.0676 | 1.3170 | 0.6622 | 2.2406 | 0.0690 | 0.2052 | 0.0434 |

22,564 | 0.2193 | 0.7735 | 0.3002 | 0.1932 | 0.6699 | 0.0154 | 0.0467 | 0.0098 |

88,036 | 0.0554 | 0.1939 | 0.0750 | 0.0472 | 0.2187 | 0.0038 | 0.0118 | 0.0023 |

349,660 | 0.0140 | 0.0494 | 0.0192 | 0.0119 | 0.0494 | 0.0010 | 0.0031 | 0.0006 |

1,409,884 | 0.0035 | 0.0121 | 0.0047 | 0.0029 | 0.0152 | 0.0002 | 0.0008 | 0.0001 |

h | ||||||||

0.4129 | – | – | – | – | – | – | – | – |

0.1940 | 1.9507 | 1.8241 | 1.9580 | 1.6310 | 1.5987 | 1.9811 | 1.9595 | 1.9687 |

0.0995 | 2.0591 | 2.0712 | 2.0759 | 2.1084 | 1.6752 | 2.1123 | 2.0631 | 2.1848 |

0.0527 | 2.1678 | 2.1538 | 2.1490 | 2.1666 | 2.3443 | 2.0727 | 2.1265 | 2.1000 |

0.0311 | 2.6452 | 2.6677 | 2.6506 | 2.6598 | 2.2355 | 2.6761 | 2.6587 | 2.6447 |

### 6.2 Example 2: Three-Dimensional Smooth Exact Solution

In our second example, we consider

With respect to boundary conditions, we impose Dirichlet conditions on

with

and

Concerning the stabilization parameters, we take them again as in Section 3.3,
but this time with

Part of the solution is displayed in Figure 2,
and a convergence history for a set of quasi-uniform mesh refinements is provided
in Table 3, thus showing also that, having the problem a smooth exact solution, this
fully-mixed finite element method converges optimally with order

Finite Element: | ||||||||

DOF | ||||||||

1,117 | 0.0219 | 0.2559 | 0.0420 | 0.0249 | 0.0262 | 0.7239 | 17.6230 | 1.2979 |

8,181 | 0.0128 | 0.1367 | 0.0265 | 0.0176 | 0.0196 | 0.4249 | 8.0953 | 0.6128 |

62,821 | 0.0079 | 0.0700 | 0.0140 | 0.0097 | 0.0132 | 0.2240 | 4.3469 | 0.3044 |

492,741 | 0.0042 | 0.0351 | 0.0071 | 0.0047 | 0.0077 | 0.1137 | 2.1971 | 0.1505 |

3,903,877 | 0.0022 | 0.0175 | 0.0035 | 0.0022 | 0.0040 | 0.0571 | 1.1017 | 0.0750 |

h | ||||||||

0.7071 | – | – | – | – | – | – | – | – |

0.3536 | 0.7778 | 0.9044 | 0.6636 | 0.4972 | 0.4220 | 0.7689 | 1.1223 | 1.0828 |

0.1768 | 0.7024 | 0.9661 | 0.9205 | 0.8660 | 0.5634 | 0.9235 | 0.8971 | 1.0094 |

0.0884 | 0.9094 | 0.9951 | 0.9871 | 1.0518 | 0.7899 | 0.9788 | 0.9844 | 1.0165 |

0.0442 | 0.9513 | 1.0018 | 1.0000 | 1.0742 | 0.9191 | 0.9943 | 0.9958 | 1.0044 |

**Funding statement: **This research 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; and by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción.

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**Received:**2018-07-26

**Revised:**2019-03-07

**Accepted:**2019-03-14

**Published Online:**2019-04-09

**Published in Print:**2020-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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