1 Introduction

In meteorological applications the following system of compressible Navier–Stokes equations governing the motion of viscous Newtonian fluid is often used, see, e.g., [1, 6, 12, 14],

$$\begin{aligned}&\partial _t\varrho + \textrm{div}_{\varvec{x}}(\varrho \varvec{u}) = 0 \end{aligned}$$
$$\begin{aligned}&\partial _t(\varrho \varvec{u}) + \textrm{div}_{\varvec{x}}(\varrho \varvec{u}\otimes \varvec{u}) + \nabla _{\varvec{x}} p(\varrho \theta ) = \textrm{div}_{\varvec{x}}({\mathbb {S}}(\nabla _{\varvec{x}}\varvec{u})) \end{aligned}$$
$$\begin{aligned}&\partial _t(\varrho \theta ) + \textrm{div}_{\varvec{x}}(\varrho \theta \varvec{u}) = 0, \end{aligned}$$

where \(\varrho \ge 0\), \(\varvec{u}\), and \(\theta \ge 0\), denote the fluid density, velocity, and potential temperature, respectively. The viscous stress tensor \({\mathbb {S}}(\nabla _{\varvec{x}}\varvec{u})\) is determined by the stipulation

$$\begin{aligned} {\mathbb {S}}(\nabla _{\varvec{x}}\varvec{u}) = \mu \left( \nabla _{\varvec{x}}\varvec{u}+ (\nabla _{\varvec{x}}\varvec{u})^T-\frac{2}{d}\,\textrm{div}_{\varvec{x}}(\varvec{u}){\mathbb {I}}\right) +\lambda \,\textrm{div}_{\varvec{x}}(\varvec{u})\,{\mathbb {I}}\,, \end{aligned}$$

where d is the space dimension, here \(d=2,3\), and the viscosity constants \(\mu \) and \(\lambda \) satisfy \( \mu > 0 \) and \(\lambda \ge -\frac{2}{d}\,\mu \), respectively. The state equation for the pressure p reads

$$\begin{aligned} p(\varrho \theta ) = a(\varrho \theta )^\gamma \,, \qquad a = \textrm{const}. >0\,, \end{aligned}$$

where \(\gamma >1\) is the so-called adiabatic index. System (1.1)–(1.3) is solved on \((0,T) \times \Omega \), where \(T>0\) is a given time and \(\Omega \subset {\mathbb {R}}^d\) a bounded domain. It is accompanied with the initial data

$$\begin{aligned} \varrho (0,\cdot ) = \varrho _0 \,, \qquad \theta (0,\cdot ) = \theta _{0}\,, \qquad \varvec{u}(0,\cdot ) = \varvec{u}_0\,, \end{aligned}$$

and no-slip boundary conditions

$$\begin{aligned} \varvec{u}|_{[0,T]\times \partial \Omega } = {\textbf{0}}\,. \end{aligned}$$

In the sequel, we shall call system (1.1)–(1.5) the Navier–Stokes system with potential temperature transport. For a brief overview of analytical results for this system we refer to our recent paper [15]. It is to be pointed out that the existence of global-in-time weak solutions to (1.1)–(1.5) is available in three space dimensions only for \(\gamma \ge 9/5\), see Maltese et al. [17, Theorem 1 with \({\mathcal {T}}(s)=s^\gamma \)]. However, physically relevant values of the adiabatic index \(\gamma \) lie in the interval (1, 5/3] for \(d=3\). This drawback motivated our recent paper [15], where we have identified a larger class of generalized solutions, dissipative measure-valued (DMV) solutions, to the Navier–Stokes system with potential temperature transport. Analyzing the convergence of a suitable numerical scheme, the mixed finite element–finite volume method, we have proved global-in-time existence of DMV solutions for all adiabatic indices \(\gamma > 1\) for \(d=2,3.\)

The first goal of the present paper is to show that the strong solutions to the Navier–Stokes system with potential temperature transport are stable in the class of DMV solutions. To this end we establish a DMV-strong uniqueness principle. This result states that the DMV and strong solutions emanating from the same initial data coincide. The key concept for the proof of this principle is the relative energy. This approach for proving weak-strong uniqueness is not new; see, e.g., [3], where DMV-strong uniqueness is proven for the Navier–Stokes system, and [7, Chapter 6], where DMV-strong uniqueness is proven for the barotropic Euler system, the complete Euler system, and the Navier–Stokes system. However, till now the weak-strong uniqueness principle was not available for the Navier–Stokes equations with potential temperature transport (1.1)–(1.5). The essential difficulty lies in the pressure law that only depends on the total potential temperature \(\varrho \theta \), without any independent control of the density \(\varrho \). To cure this problem, we will rewrite the pressure as a function of the density and total physical entropy. This allows us to separate the effects of the density and potential temperature in the derivation of the relative energy and finally to show the DMV-strong uniqueness principle.

The second goal is to derive a priori error estimates for the finite element–finite volume method proposed in [15]. To this end, we assume that the strong solution exists and apply a relative energy inequality and a consistency formulation for the numerical method. Such an approach has already been applied successfully to the compressible Navier–Stokes equations, see Kwon and Novotný [13], and to the compressible Euler system, see [16]. However, in those works, the approximation of a sufficiently smooth domain \(\Omega \subset {\mathbb {R}}^d\) by a sequence of polygonal approximations \(\Omega _h\subset {\mathbb {R}}^d\), \(h\downarrow 0\), was not considered. In the present paper, novel consistency estimates are presented that allow to compare a strong solution on a smooth domain \(\Omega \) with numerical solutions computed on polygonal domains \(\Omega _h\), \(\Omega \subset \Omega _h\). Here, we only assume that \(\textrm{dist}(\varvec{x},\partial \Omega )={\mathcal {O}}(h)\) for all \(\varvec{x}\in \partial \Omega _h,\) see also Feireisl et al. [4, 8] for related results for the compressible Navier–Stokes equations on general domains under slightly more restrictive assumptions.

The paper is organized as follows: In Sect. 2, we briefly repeat the relevant notation and our definition of DMV solutions to Navier–Stokes system with potential temperature transport proposed in [15]. Section 3 is devoted to the proof of the DMV-strong uniqueness principle. Further, the error estimates are derived in Sect. 4 where we also present some numerical results.

2 DMV Solutions

We start by introducing the pressure potential \(P:[0,\infty )\rightarrow {\mathbb {R}}\) as

$$\begin{aligned} P(z) = \frac{a}{\gamma -1}\,z^\gamma \,. \end{aligned}$$

In what follows we write \(\Omega _t = (0,t)\times \Omega \) whenever \(t>0.\) If \({\mathcal {V}}=\{{\mathcal {V}}_{(t,\varvec{x})}\}_{(t,\varvec{x})\,\in \,\Omega _T}\) is a space-time parametrized probability measure acting on \({\mathbb {R}}^{d+2}\), we write

whenever \(g\in C({\mathbb {R}}^{d+2})\). In particular, we tend to write out the function g in terms of the integration variables : if, for example, , then we also write

We recall the definition of dissipative measure-valued solutions to the Navier–Stokes system with potential temperature transport (1.1)–(1.5) from [15].

Definition 2.1

(DMV solutions, [15, Definition 2.1]). A parametrized probability measure \({\mathcal {V}}=\{{\mathcal {V}}_{(t,\varvec{x})}\}_{(t,\varvec{x})\,\in \,\Omega _T}\) that satisfies

Footnote 1and for which there exists a constant \(c_\star >0\) such that


is called a dissipative measure-valued (DMV) solution to the Navier–Stokes system with potential temperature transport (1.1)–(1.5) with initial and boundary conditions (1.6) and (1.7) if it satisfies:

  • Energy inequality

    and the integral inequality


    holds for a.a. \(\tau \in (0,T)\) with the energy concentration defect

    and the dissipation defect

    $$\begin{aligned} {\mathfrak {D}}\in {\mathcal {M}}^+(\,\overline{\Omega _T})\,; \end{aligned}$$
  • Continuity equation

    and the integral identity


    holds for all \(\tau \in [0,T]\) and all \(\varphi \in W^{1,\infty }(\Omega _T)\)Footnote 2;

  • Momentum equation

    and the integral identity


    holds for all \(\tau \in [0,T]\) and all \(\varvec{\varphi }\in C^{1}(\,\overline{\Omega _T})^d\) satisfying \(\varvec{\varphi }|_{[0,T]\times \partial \Omega }=\varvec{0}\), where the Reynolds concentration defect fulfills

    Footnote 3

  • Potential temperature equation

    and the integral identity


    holds for all \(\tau \in [0,T]\) and all \(\varphi \in W^{1,\infty }(\Omega _T)\);

  • Entropy inequality

    and for any \(\psi \in W^{1,\infty }(\Omega _T)\), \(\psi \ge 0\), the integral inequality


    is satisfied for a.a. \(\tau \in (0,T)\);

  • Poincaré’s inequality

    there exists a constant \(C_P>0\) such that


for a.a. .

Remark 2.2

As we shall see in the next section, the entropy inequality (2.7) and Poincaré’s inequality (2.8) included in the definition of DMV solutions to the Navier–Stokes system with potential temperature transport are fundamental to guarantee DMV-strong uniqueness.

3 DMV-Strong Uniqueness

The aim of this section is to derive a DMV-strong uniqueness principle for our measure-valued solutions. For this purpose, we rely on the concept of relative energy. We introduce the (physical) entropy S as

$$\begin{aligned} S = S(\varrho ,\theta ) = {\frac{\gamma }{\gamma -1}\,\varrho \ln (\theta )} \end{aligned}$$

and realize that the pressure \(p=a(\varrho \theta )^\gamma \) can be rewritten with respect to \(\varrho \), S as

$$\begin{aligned} p(\varrho ,S) = { \mathbb {1}_{\{\varrho \,>\,0\}}\,a\varrho ^\gamma \exp \left( (\gamma -1)\,\dfrac{S}{\varrho }\right) }\,. \end{aligned}$$

We proceed by defining the relative energy between a triplet of arbitrary functions \((\varrho ,\theta ,\varvec{u})\) belonging to a regularity class

$$\begin{aligned} \varrho ,\theta \in C^{1}(\overline{\Omega _T})\,, \quad \varrho ,\theta >0\,, \quad \varvec{u}\in C^{1}(\overline{\Omega _T}) \cap L^2(0,T; W^{2,\infty }(\Omega ))\,, \quad \varvec{u}|_{[0,T]\times \partial \Omega }=\varvec{0}, \end{aligned}$$

and a DMV solution \({\mathcal {V}}\) to the Navier–Stokes system with potential temperature transport (1.1)–(1.5) as


where \(P(\varrho ,S)=\frac{1}{\gamma -1}\,p(\varrho ,S)\) is the pressure potential expressed in terms of \(\varrho \) and S, \(S=S(\varrho ,\theta )\), and .

Remark 3.1

We note that \(P=P(\varrho ,S)\) satisfies the following identity on \((0,\infty )\times {\mathbb {R}}\):

$$\begin{aligned} \frac{\partial P(\varrho ,S)}{\partial \varrho }\,\varrho + \frac{\partial P(\varrho ,S)}{\partial S}\,S - P(\varrho ,S) = p(\varrho ,S)\, \end{aligned}$$

We further note that we only consider the case in which are bounded from below by some constant \(c>0\) (for \(\theta \) this is reflected by (3.3) and for by (2.2)). Consequently, (3.1) makes sense. In particular, \(S(\varrho ,\theta )\) and the composition \(p(\varrho ,S(\varrho ,\theta ))\) are continuous functions of \((\varrho ,\theta )\) on \([0,\infty )\times [c,\infty )\) for every \(c>0\). In addition, we shall always construe S and as functions of \(\varrho ,\theta \) and , respectively. Accordingly, for example, .

The relative energy inequality corresponding to (3.4) reads as follows.

Lemma 3.2

(Relative energy inequality). Let \((\varrho ,\theta ,\varvec{u})\) be a triplet of test functions, cf. (3.3), and \({\mathcal {V}}\) a DMV solution to (1.1)–(1.5) in the sense of Definition 2.1. Then the relative energy defined in (3.4) satisfies the inequality


for a.a. \(\tau \in (0,T)\). Here,

$$\begin{aligned} \vartheta = \frac{1}{\gamma -1}\,\frac{\partial p(\varrho ,S)}{\partial S}=\frac{\partial P(\varrho ,S)}{\partial S} \end{aligned}$$

denotes the absolute temperature.


Using Gauss’s theorem we easily verify that


Next, using the definition of the absolute temperature, cf. (3.7), and (3.5) we deduce that


Combining (3.8) and (3.9) with (2.3)–(2.7), we obtain

In the next step, we carry out the partial derivatives on the right-hand side of the above inequality. In doing so, we take into account that (3.5) implies that for any \(w\in \{t,x_1,\dots ,x_d\}\),

$$\begin{aligned} \partial _w\,\frac{\partial P}{\partial \varrho }&= \frac{\partial ^2 P}{\partial \varrho ^2}\,\partial _w\varrho + \frac{\partial ^2 P}{\partial S\,\partial \varrho }\,\partial _w S = \left[ \frac{1}{\varrho }\frac{\partial p}{\partial \varrho }-\frac{S}{\varrho }\frac{\partial ^2 P}{\partial \varrho \,\partial S}\right] \partial _w\varrho + \left[ \frac{1}{\varrho }\frac{\partial p}{\partial S}-\frac{S}{\varrho }\frac{\partial ^2 P}{\partial S^2}\right] \partial _w S \\&= \frac{1}{\varrho }\left[ \frac{\partial p}{\partial \varrho }\,\partial _w \varrho + \frac{\partial p}{\partial S}\,\partial _w S\right] + \frac{S}{\varrho }\,\partial _w\,\frac{\partial P}{\partial S} = \frac{1}{\varrho }\left[ \frac{\partial p}{\partial \varrho }\,\partial _w \varrho + \frac{\partial p}{\partial S}\,\partial _w S\right] + \frac{S}{\varrho }\,\partial _w\vartheta \,, \end{aligned}$$

where the last equality is due to (3.7). Consequently, we get

To finish the proof of Lemma 3.2, we add and subtract the following terms on the right-hand side of the above inequality

and regroup the resulting expressions adequately. \(\square \)

From the relative energy inequality we can deduce the DMV-strong uniqueness result.

Theorem 3.3

(DMV-strong uniqueness). Let \(\gamma >1\), \(\Omega \subset {\mathbb {R}}^d\), \(d\in \{2,3\}\), be a bounded domain of class \(C^3\). Further, let \(T^* > 0 \) and \((\varrho ,\theta ,\varvec{u})\) be a strong solution to system (1.1)–(1.5) on \(\Omega _{T^*}\) belonging to the regularity class (3.3). Let \({\mathcal {V}}\) be a DMV solution in the sense of Definition 2.1 emanating from the same initial data. Then

$$\begin{aligned} {\mathfrak {E}}=0\,, \quad {\mathfrak {D}}([0,T^*)\times {\overline{\Omega }})=0\,, \quad \varvec{{\mathfrak {R}}}=\varvec{0}\,, \end{aligned}$$

and the DMV and strong solutions coincide on \([0,T^*)\), i.e.

$$\begin{aligned} {\mathcal {V}}_{(t,\varvec{x})}=\delta _{(\varrho (t,\varvec{x}),\theta (t,\varvec{x}),\varvec{u}(t,\varvec{x}))} \quad \text {for a.a. } (t,\varvec{x})\in \Omega _{T^*}. \end{aligned}$$


Plugging the strong solution \((\varrho ,\theta ,\varvec{u})\) into the relative energy inequality (3.6), we obtain


for a.a. \(\tau \in (0,T^*)\). To handle the last two integrals, we first observe that

$$\begin{aligned} \int _{0}^{\tau }\int _{{{\Omega }}}{\mathbb {S}}(\nabla _{\varvec{x}}(\varvec{u}_{\mathcal {V}}-\varvec{u})): \nabla _{\varvec{x}}(\varvec{u}_{\mathcal {V}}-\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t&= \int _{0}^{\tau }\int _{{{\Omega }}}\big [\mu |\nabla _{\varvec{x}}(\varvec{u}_{\mathcal {V}}-\varvec{u})|^2+\nu |\textrm{div}_{\varvec{x}}(\varvec{u}_{\mathcal {V}} -\varvec{u})|^2\big ]\,\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&\ge \mu \int _{0}^{\tau }\int _{{{\Omega }}}|\nabla _{\varvec{x}}(\varvec{u}_{\mathcal {V}}-\varvec{u})|^2\;\textrm{d}\varvec{x}\textrm{d}t\,. \end{aligned}$$

Next, we set

$$\begin{aligned} ({\underline{\varrho }},{\overline{\varrho }},{\underline{\theta }},{\overline{\theta }}) = \left( \inf _{(t,\varvec{x})\,\in \,\Omega _{T^*}}\{\varrho (t,\varvec{x})\}, \sup _{(t,\varvec{x})\,\in \,\Omega _{T^*}}\{\varrho (t,\varvec{x})\},\inf _{(t,\varvec{x})\,\in \,\Omega _{T^*}}\{\theta (t,\varvec{x})\}, \sup _{(t,\varvec{x})\,\in \,\Omega _{T^*}}\{\theta (t,\varvec{x})\}\right) \end{aligned}$$

and apply Lemma A.1 to find constants \(c_1,c_2,c_3>0\) that only depend on \({\underline{\varrho }}\), \({\overline{\varrho }}\), \({\underline{\theta }}\), \({\overline{\theta }}\), \(c_\star \), and \(\gamma \), and corresponding sets

such that


Seeing that

as well as

we may use (3.12) to deduce


We proceed by observing that


for all \(\alpha >0\), where here and in the sequel the constant hidden in “\(\lesssim \)” does not depend on \(\alpha \). Together with (3.12) and Poincaré’s inequality (2.8), these observations yield


Finally, combining (3.10), (3.11), (3.13), and (3.14), we arrive at

$$\begin{aligned}&\left[ \,\int _{{{\Omega }}}E({\mathcal {V}}|\varrho ,\theta , \varvec{u})(t,\cdot )\;\textrm{d}\varvec{x}\right] _{t = 0}^{t = \tau } + \int _{\,{\overline{\Omega }}}\,\textrm{d}{\mathfrak {E}}(\tau ) + \int _{\,\overline{\Omega _\tau }}\textrm{d}{\mathfrak {D}} + \mu \int _{0}^{\tau }\int _{{{\Omega }}}|\nabla _{\varvec{x}}(\varvec{u}_{\mathcal {V}}-\varvec{u})|^2\;\textrm{d}\varvec{x}\textrm{d}t\\&\quad \lesssim (1+\alpha ^{-1}) \int _{0}^{\tau }\int _{{{\Omega }}}E({\mathcal {V}} |\varrho ,\theta ,\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ (1+\alpha )\int _{0}^{\tau }\int _{\, {\overline{\Omega }}}\,\textrm{d}{\mathfrak {E}}(t)\textrm{d}t\\&\qquad +\alpha \left( \,\int _{0}^{\tau }\int _{{{\Omega }}}|\nabla _{\varvec{x}}(\varvec{u}_{{\mathcal {V}}}-\varvec{u})|^2\;\textrm{d}\varvec{x}\textrm{d}t+ \int _{\,\overline{\Omega _\tau }}\textrm{d}{\mathfrak {D}}\right) \end{aligned}$$

for a.a. \(\tau \in (0,T^*)\) and all \(\alpha >0\). In particular, there exists a constant \(C>0\) such that

$$\begin{aligned}&\left[ \,\int _{{{\Omega }}}E({\mathcal {V}}|\varrho ,\theta ,\varvec{u})(t,\cdot )\;\textrm{d}\varvec{x}\right] _{t = 0}^{t = \tau } + \int _{\,{\overline{\Omega }}}\,\textrm{d}{\mathfrak {E}}(\tau ) + \int _{\,\overline{\Omega _\tau }}\textrm{d}{\mathfrak {D}} \\&\quad \le C\left( \,\int _{0}^{\tau }\int _{{{\Omega }}}E({\mathcal {V}}|\varrho ,\theta ,\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+\int _{0}^{\tau }\int _{\,{\overline{\Omega }}}\,\textrm{d}{\mathfrak {E}}(t)\textrm{d}t+\int _{0}^{\tau }\int _{\,\overline{\Omega _t}}\textrm{d}{\mathfrak {D}}\textrm{d}t\right) \end{aligned}$$

for a.a. \(\tau \in (0,T^*)\). Consequently, the desired result follows from Gronwall’s lemma. \(\square \)

Remark 3.4

The local existence of strong solutions to (1.1)–(1.5) for the Cauchy problem (i.e. \(\Omega ={\mathbb {R}}^d\)) is guaranteed by [11, Theorem 2.9] and the global existence for small initial data by [11, Theorem 3.6]. These results apply to a class of systems of hyperbolic-parabolic composite type. The local existence result just mentioned was generalized in [19]. We expect that these results can be transferred to the initial-boundary value problem considered here provided \(\Omega \) is of class \(C^3\) and the initial data satisfy suitable compatibility conditions. This can be an interesting topic for future studies.

4 Error Estimates for a Numerical Scheme

In our recent paper [15], we have introduced a mixed finite element-finite volume (FE-FV) numerical method and showed that in a suitable (weak) sense its solutions converge to a DMV solution to the Navier–Stokes equations with potential temperature transport (1.1)–(1.5). Moreover, we proved that if a strong solution exists, then the numerical solutions converge strongly to this strong solution, cf. [15, Theorem 6.1].

The ultimate goal of this section is to strengthen the just mentioned result and derive a priori error estimates for the finite element-finite volume method applying the relative energy method.

The section is organized as follows: In Sect. 4.1, we formulate minimal regularity assumptions required for the strong solution and the initial data. Sections 4.2 and 4.3 are devoted to the recapitulation of the numerical scheme presented in [15] and its properties. In Sect. 4.4, we present a novel consistency formulation taking the approximation of a smooth domain \(\Omega \) by a sequence of polygonal computational domains \(\Omega _h\), \(h \downarrow 0,\) into account. The desired error estimates are presented in Sect. 4.5. We finish this section by presenting some numerical results illustrating the convergence of the scheme.

4.1 Regularity Class for the Strong Solution and the Initial Data

We will consider strong solutions \((\varrho ,\theta ,\varvec{u})\) to (1.1)–(1.5) that belong to the regularity class


Accordingly, the initial data satisfy

$$\begin{aligned} \varrho _0,\theta _{0}\in C^{1}({\overline{\Omega }})\,, \quad \varrho _0,\theta _{0}>0\,, \quad \varvec{u}_0\in C^{1}({\overline{\Omega }})\,, \quad \varvec{u}_0|_{\partial \Omega }=\varvec{0}\,. \end{aligned}$$

For functions such as \(\varrho \) in (4.1) and \(\varrho _0\) in (4.2), we further introduce the following notation:

$$\begin{aligned} (\varrho _\star ,\varrho ^\star ,(\varrho _0)_\star ,(\varrho _0)^\star ) = \left( \inf _{(t,\varvec{x})\,\in \,\Omega _T}\{\varrho (t,\varvec{x})\}, \sup _{(t,\varvec{x})\,\in \,\Omega _T}\{\varrho (t,\varvec{x})\},\inf _{\varvec{x}\,\in \,\Omega }\{\varrho _0(\varvec{x})\},\sup _{\varvec{x}\,\in \,\Omega }\{\varrho _0(\varvec{x})\}\right) . \end{aligned}$$

In addition, we consider the initial data \((\varrho _0,\theta _{0},\varvec{u}_0)\) to be extended by \(((\varrho _0)_\star ,(\theta _{0})_\star ,\varvec{0})\) outside \({\overline{\Omega }}\).

4.2 Mixed Finite Element-Finite Volume Method

We recall the mixed FE-FV numerical method introduced in [15]Footnote 4, see also [5, Chapter 7].

4.2.1 Spatial Discretization

Let \(H\in (0,1)\). The spatial domain \(\Omega \subset {\mathbb {R}}^d\) is approximated by a family \(\{\Omega _h\}_{h\,\in \,(0,H]}\) that is connected to a family of (finite) meshes \(({\mathcal {T}}_h)_{h\,\in \,(0,H]}\) via the constraint

$$\begin{aligned} \overline{\Omega _h} = \bigcup _{K\,\in \,{\mathcal {T}}_h} K \qquad \text {for all }h\in (0,H]. \end{aligned}$$

We assume that the following conditions are satisfied:

  • There is a constant \(D>0\) such that

    $$\begin{aligned} \Omega \subset \Omega _h\subset \big \{\varvec{x}\in {\mathbb {R}}^d\,\big |\, \textrm{dist}(\varvec{x},{\overline{\Omega }})<Dh\big \} \quad \text {for all }h\in (0,H]; \end{aligned}$$
  • Each element K of a mesh \({\mathcal {T}}_h\) is a d-simplex that can be written as

    $$\begin{aligned} K = h{\mathbb {A}}_K (K_{\textrm{ref}}) + {\textbf{a}}_K\,, \qquad {\mathbb {A}}_K\in {\mathbb {R}}^{d\times d}\,, \qquad {\textbf{a}}_K\in {\mathbb {R}}^d\,, \end{aligned}$$

    where the reference element \(K_\textrm{ref}\) is the convex hull of the zero vector \(\varvec{0}\in {\mathbb {R}}^d\) and the standard unit vectors \({\textbf{e}}_{1},\dots ,{\textbf{e}}_{d}\in {\mathbb {R}}^d\), i.e., \(K_{\textrm{ref}} = \textrm{conv}\{{\textbf{0}},{\textbf{e}}_{1},\dots ,{\textbf{e}}_{d}\}\,\);

  • There exist constants \(C>c>0\) such that \(\textrm{spectrum}({\mathbb {A}}_K^T{\mathbb {A}}_K) \subset [c,C]\) for all \(K\in {\bigcup _{\,h\,\in \,(0,H]}}{\mathcal {T}}_h\) ;

  • The intersection of two distinct elements \(K_1,K_2\) of a mesh \({\mathcal {T}}_h\) is either empty, a common vertex, a common edge, or (in the case \(d=3\)) a common face.

The symbol \({\mathcal {E}}_{h}\) denotes the set of all faces, \(d=3\), or all edges, \(d=2\), in the mesh \({\mathcal {T}}_h\). \({\mathcal {E}}_{h,\textrm{ext}}\) and \({\mathcal {E}}_{h,\textrm{int}}\) stand for the sets of exterior and interior faces, respectively, i.e.,

$$\begin{aligned} {\mathcal {E}}_{h,\textrm{ext}}= \big \{\sigma \in {\mathcal {E}}_{h}\,\big |\,\sigma \subset \partial \Omega _h\big \} \qquad \text {and} \qquad {\mathcal {E}}_{h,\textrm{int}}= {\mathcal {E}}_{h}\backslash {\mathcal {E}}_{h,\textrm{ext}}\,. \end{aligned}$$

Moreover, for \(K\in {\mathcal {T}}_h\), we put \({\mathcal {E}}_h(K) = \big \{\sigma \in {\mathcal {E}}_h\,\big |\,\sigma \subset K\big \}\) and \({\mathcal {E}}_{h,z}(K) = \big \{\sigma \in {\mathcal {E}}_{h,z}\,\big |\,\sigma \subset K\big \}\), where \(z\in \{\textrm{int},\textrm{ext}\}\). In connection with these sets, we shall use the abbreviations

$$\begin{aligned} \int _{{\mathcal {E}}_{h,\textrm{int}}} \equiv \sum _{\sigma \,\in \,{\mathcal {E}}_{h,\textrm{int}}}\int _{{{\sigma }}}\qquad \text {and} \qquad \int _{{\mathcal {E}}_{h}(K)} \equiv \sum _{K \,\in \, {\mathcal {T}}_h}\sum _{\sigma \,\in \,{\mathcal {E}}_h(K)}\int _{{{\sigma }}}\,. \end{aligned}$$

Each face \(\sigma \in {\mathcal {E}}_{h}\) is equipped with a unit vector \(\varvec{n}_\sigma \) that is determined as follows: We fix an arbitrary element \(K_\sigma \in {\mathcal {T}}_h\) such that \(\sigma \in {\mathcal {E}}_h(K_\sigma )\) and set \(\varvec{n}_\sigma = \varvec{n}_{K_\sigma }(\varvec{x}_\sigma )\). Here, \(\varvec{x}_\sigma \) denotes the center of mass of \(\sigma \) and \(\varvec{n}_{K_\sigma }(\varvec{x}_\sigma )\) is the outward-pointing unit normal vector to the element \(K_\sigma \) at \(\varvec{x}_\sigma \). Finally, it will be convenient to write \(A \lesssim B\) whenever there is an h-independent constant \(c>0\) such that \(A \le cB\) and \(A \approx B\) whenever \(A\lesssim B\) and \(B\lesssim A\).

4.2.2 Function Spaces and Projection Operators

The space of piecewise constant functions is denoted by

$$\begin{aligned} Q_h = \big \{v\in L^2(\Omega _h)\,\big |\, v|_K\in P_{{{0}}}(K)\text { for all }K\in {\mathcal {T}}_h\big \}. \end{aligned}$$

Footnote 5For \(v\in Q_h\) and \(K\in {\mathcal {T}}_h\) we set \(v_K = v(\varvec{x}_K)\), where \(\varvec{x}_K\) denotes the center of mass of K. The projection \(\Pi _{Q,h}\equiv \overline{\;\cdot \;}:L^2(\Omega _h)\rightarrow Q_h\) associated with \(Q_h\) is characterized by

$$\begin{aligned} (\Pi _{Q,h}v)\big |_K \equiv {\overline{v}}|_K \equiv \frac{1}{|K|}\int _{{{K}}}v\;\textrm{d}{\textbf{y}}\quad \text {for all }K\in {\mathcal {T}}_h\,. \end{aligned}$$

The Crouzeix-Raviart finite element spaces are denoted by

$$\begin{aligned} V_h&= \left\{ v\in L^2(\Omega _h)\left| \,\begin{array}{c} v|_K\in P_{{{1}}}(K)\text { for all }K\in {\mathcal {T}}_h\text { and} \\ {\displaystyle \int _{{{\sigma }}}\,\lim _{\delta \,\rightarrow \,0^+}}\big ( v(\varvec{x}-\delta \varvec{n}_\sigma )-v(\varvec{x}+\delta \varvec{n}_\sigma )\big )\,\textrm{d}S_{\varvec{x}}= 0 \quad \text {for all }\sigma \in {\mathcal {E}}_{h,\textrm{int}}\end{array}\right. \right\} , \\ V_{0,h}&= \left\{ v\in V_h\left| \;\int _{{{\sigma }}}\, \lim _{\delta \,\rightarrow \,0^+}v(\varvec{x}-\delta \varvec{n}_\sigma )\;\textrm{d}S_{\varvec{x}}= 0 \quad \text {for all }\sigma \in {\mathcal {E}}_{h,\textrm{ext}}\right. \right\} \,. \end{aligned}$$

With these spaces we associate the projections \(\Pi _{V,h}:W^{1,2}(\Omega _h)\rightarrow V_h\), \(\Pi _{V,h}^0:W^{1,2}(\Omega _h)\rightarrow V_{0,h}\) that are determined by

$$\begin{aligned} \int _{{{\sigma }}}\Pi _{V,h}v\;\textrm{d}S_{\varvec{x}}= \int _{{{\sigma }}}v\;\textrm{d}S_{\varvec{x}}\quad \text {for all } \sigma \in {\mathcal {E}}_{h}, \qquad \int _{{{\sigma }}}\Pi _{V,h}^0v\;\textrm{d}S_{\varvec{x}}= \int _{{{\sigma }}}v\;\textrm{d}S_{\varvec{x}}\quad \text {for all }\sigma \in {\mathcal {E}}_{h,\textrm{int}}, \end{aligned}$$

respectively. Additionally, we agree on the notation

$$\begin{aligned} Q_h^+= & {} \big \{v\in Q_h\,\big |\,v|_K > 0 \;\; \text {for all } K\in {\mathcal {T}}_h\big \}\,, \qquad Q_h^{0,+} = \big \{v\in Q_h\,\big |\,v|_K \ge 0 \;\; \text {for all } K\in {\mathcal {T}}_h\big \}\,, \\ \varvec{Q}_h= & {} (Q_h)^d\,, \qquad \varvec{V}_h = (V_h)^d\,, \qquad \text {and} \qquad \varvec{V}_{0,h} = (V_{0,h})^d\,. \end{aligned}$$

4.2.3 Mesh-Related Operators

Next, we recall the necessary mesh-related operators. We start by repeating the definitions of the discrete counterparts of the differential operators \(\nabla _{\varvec{x}}\) and \(\textrm{div}_{\varvec{x}}\). They are determined by the stipulations

$$\begin{aligned} \begin{aligned}&(\nabla _{h}\varvec{v})|_K = \nabla _{\varvec{x}}(\varvec{v}|_K) \quad \text {for all } \varvec{v}\in (V_h\cup \varvec{V}_h)\cup (W^{1,1}(\Omega _h)\cup W^{1,1}(\Omega _h)^d)\text { and all }K\in {\mathcal {T}}_h \\&\quad \text {and} \quad \textrm{div}_{{{h}}}(\varvec{v})|_{K} = \textrm{div}_{\varvec{x}}(\varvec{v}|_K) \quad \text {for all }\varvec{v}\in \varvec{V}_h\cup W^{1,1}(\Omega _h)^d\text { and all }K\in {\mathcal {T}}_h, \end{aligned} \end{aligned}$$

respectively. We continue by recalling the trace operators. For arbitrary \(\sigma \in {\mathcal {E}}_{h}\), \(\varvec{x}\in \sigma \), and

$$\begin{aligned} \varvec{v}\in (Q_h\cup \varvec{Q}_h)\cup (V_h\cup \varvec{V}_h)\cup (C(\overline{\Omega _h})\cup C(\overline{\Omega _h})^d) \end{aligned}$$

we set

$$\begin{aligned} \varvec{v}^{\,\textrm{in},\,\sigma }(\varvec{x})= & {} \lim _{\delta \,\rightarrow \,0^+}\varvec{v}(\varvec{x}-\delta \varvec{n}_\sigma )\,, \qquad \varvec{v}^{\,\textrm{out},\,\sigma }(\varvec{x}) =\left\{ \begin{array}{ll} {\displaystyle \lim _{\delta \,\rightarrow \,0^+}\varvec{v}(\varvec{x}+\delta \varvec{n}_\sigma )} &{} \text {if }\sigma \in {\mathcal {E}}_{h,\textrm{int}},\\ \varvec{0} &{} \text {else} \end{array}\right. \,, \\ \llbracket \varvec{v}\rrbracket _\sigma= & {} \varvec{v}^{\,\textrm{out},\,\sigma }-\varvec{v}^{\,\textrm{in},\,\sigma }\,, \qquad \{\varvec{v}\}_\sigma = \dfrac{\varvec{v}^{\,\textrm{out},\,\sigma }+\varvec{v}^{\,\textrm{in},\,\sigma }}{2} \qquad \text {and} \qquad \langle \varvec{v} \rangle _\sigma = \frac{1}{|\sigma |}\int _{{{\sigma }}}\varvec{v}^{\,\textrm{in},\,\sigma }\;\textrm{d}S_{\varvec{x}}\,. \end{aligned}$$

The convective terms shall be approximated by means of a dissipative upwind operator. For \(\sigma \in {\mathcal {E}}_{h}\), \(\varvec{v}\in \varvec{V}_{0,h}\), and \(\varvec{r}\in Q_h\cup \varvec{Q}_h\) we put

$$\begin{aligned} \textrm{Up}\left[ \varvec{r},\varvec{v}\right] _{\sigma }&= \varvec{r}^{\,\textrm{out},\,\sigma } \left[ \langle \varvec{v}\cdot \varvec{n}_\sigma \rangle _\sigma \right] ^- + \varvec{r}^{\,\textrm{in} ,\,\sigma }\left[ \langle \varvec{v}\cdot \varvec{n}_\sigma \rangle _\sigma \right] ^+\,, \\ F_h^{\,\textrm{up}}\left[ \varvec{r},\varvec{v}\right] _{\sigma }&= \textrm{Up}\left[ \varvec{r},\varvec{v}\right] _{\sigma } -\frac{h^\varepsilon }{2}\,\llbracket \varvec{r}\rrbracket _\sigma = \{\varvec{r}\}_\sigma \langle \varvec{v}\cdot \varvec{n}_\sigma \rangle _\sigma -\frac{1}{2}\, \llbracket \varvec{r}\rrbracket _\sigma \big (h^\varepsilon +|\langle \varvec{v}\cdot \varvec{n}_\sigma \rangle _\sigma |\big )\,, \end{aligned}$$

where \(\varepsilon >0\) is a given constant, \([x]^+ = \max \{x,0\}\) and \([x]^- = \min \{x,0\}\).

Remark 4.1

As in [15], we tend to omit parts of the subscripts and superscripts of the operators defined in Sects. 4.2.2 and 4.2.3 if no confusion arises. This includes the letters h and \(\sigma \) as well as the word in.

4.2.4 Time Discretization

To approximate the time derivatives, we employ the backward Euler method. Consequently, the discrete time derivative \(D_{t}\) is given by

$$\begin{aligned} D_{t}\varvec{s}^k_h = \frac{\varvec{s}^k_h-\varvec{s}^{k-1}_h}{\Delta t}\,, \end{aligned}$$

where \(\Delta t>0\) is a given time step and \(\varvec{s}^{k-1}_h\) and \(\varvec{s}^k_h\) are the numerical solutions at the time levels \(t_{k-1}=(k-1)\Delta t\) and \(t_k=k\Delta t\), respectively. For the sake of simplicity, we assume that \(\Delta t\) is constant and that there is a number \(N_T\in {\mathbb {N}}\) such that \(N_T\Delta t= T\).

4.2.5 Numerical Scheme

The mixed FE-FV method introduced in [15, Definition 3.2] reads as follows.

figure a

4.2.6 Discrete Initial Data

The initial data for the mixed FE-FV method (4.5)–(4.7) are determined as follows:

$$\begin{aligned} \varrho _h^0 = \Pi _{Q}\varrho _0\,, \qquad \theta _{h}^{0}= \Pi _{Q}\theta _{0}\qquad \text {and} \qquad \varvec{u}_h^0 = \Pi _{V}\varvec{u}_0\,. \end{aligned}$$

As a consequence of this stipulation, we observe that \((\varrho _h^0,\theta _{h}^{0},\varvec{u}_h^0)\in Q_h^+\times Q_h^+\times \varvec{V}_{0,h}\) with

$$\begin{aligned} 0<(\varrho _0)_\star \le \varrho _h^0 \le (\varrho _0)^\star \qquad \text {and} \qquad 0<(\theta _{0})_\star \le \theta _{h}^{0}\le (\theta _{0})^\star \,. \end{aligned}$$

4.3 Discrete Energy and Entropy Inequalities

The solvability of the FE-FV method (4.5)–(4.7) is guaranteed by [15, Lemma 3.4]. In particular, it follows from a combination of this lemma with (4.9) that for every \(k\in {\mathbb {N}}_0\)


In addition, it turns out that the numerical solutions satisfy an energy balance and an entropy inequality that read as follows:

figure b
figure c

Given a solution \((\varrho _h^k,\theta _{h}^{k},\varvec{u}_h^k)_{k\,\in \,{\mathbb {N}}}\subset Q_h^+\times Q_h^+\times \varvec{V}_{0,h}\) to the FE-FV method (4.5)–(4.7) starting from the initial data (4.8), we define the functions \(\varrho _h^{-},\varrho _h,\theta _{h}:{\mathbb {R}}\times \Omega _h\rightarrow (0,\infty )\), \(\varvec{u}_h:{\mathbb {R}}\times \Omega _h\rightarrow {\mathbb {R}}^d\) that are piecewise constant in time by setting

$$\begin{aligned} (\varrho _h^{-},\varrho _h,\theta _{h},\varvec{u}_h)(t,\cdot )&= \left\{ \begin{array}{ll} (\varrho _h^{k-1},\varrho _h^k,\theta _{h}^{k},\varvec{u}_h^k) &{} \quad \text {if }t\in (t_{k-1},t_k]\text { for some }k\in {\mathbb {N}}\text { and} \\ (\varrho _h^0,\varrho _h^0,\theta _{h}^{0},\varvec{u}_h^0) &{} \quad \text {if }t\le 0. \end{array} \right. \end{aligned}$$

In addition, we introduce the functions \(S_h,E_h:{\mathbb {R}}\times \Omega _h\rightarrow {\mathbb {R}}\) via the stipulations

$$\begin{aligned} S_h = \frac{\gamma }{\gamma -1}\,\varrho _h\ln (\theta _{h})\,, \qquad E_h=\frac{1}{2}\,\varrho _h|\overline{\varvec{u}_h}|^2+P(\varrho _h,S_h) +h^\delta \varrho _h^2\big [1+\theta _{h}^2\big ]\,. \end{aligned}$$

Next, let us state two consequences of the discrete energy balance (4.11).

4.3.1 Stability Estimates

From (4.11) we obtain the subsequent energy estimates (cf. [15, Corollary 4.4]):


where \(q\in [1,\infty )\) if \(d=2\) and \(q\in [1,6]\) if \(d=3\).

Remark 4.2

Note that the proof of [15, Corollary 4.4] can be extended to include the estimates in (4.23). In addition, the different way of approximating the spatial domain \(\Omega \) (we now have \(\Omega \subset \Omega _h\) instead of \(\Omega _h\subset \Omega \)) requires some minor and straightforward modifications. We leave the details to the interested reader.

Moreover, for further application it is convenient to observe that the energy balance (4.11) provides us with the following

figure d

4.4 Consistency

We proceed by stating a suitable consistency formulation of the numerical scheme (4.5)–(4.7).

Theorem 4.3

(Consistency of the FE-FV method). Let \(\beta =\min \left\{ \varepsilon -1,\frac{1-2\delta }{4}\right\} \) and \(\tau \in [0,T]\). Further, suppose \((\varrho _h,\theta _{h},\varvec{u}_h)_{h\,\in \,(0,H]}\) is a family of solutions to the FE-FV method (4.5)–(4.7) with

$$\begin{aligned} \gamma> 1\,, \qquad \Delta t\approx h\,, \qquad \varepsilon > 1 \qquad \text {and} \qquad 0<\delta <\tfrac{1}{2} \end{aligned}$$

starting from the initial data \((\varrho _h^0,\theta _{h}^{0},\varvec{u}_h^0)_{h\,\in \,(0,H]}\) defined in (4.8). Then

$$\begin{aligned} \left[ \,\int _{{{\Omega }}}(\varrho _h\varphi )(t,\cdot )\;\textrm{d}\varvec{x}\right] ^{t = \tau }_{t = 0} = \int _{0}^{\tau }\int _{{{\Omega }}}\big [\varrho _h\partial _t\varphi +\varrho _h\overline{\varvec{u}_h}\cdot \nabla _{\varvec{x}}\varphi \big ]\,\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^\beta ) \end{aligned}$$

for all \(\varphi \in C^{1}(\overline{\Omega _T})\) as \(h\downarrow 0\),

$$\begin{aligned}&\left[ \,\int _{{{\Omega }}}(\varrho _h\overline{\varvec{u}_h}\cdot \varvec{\varphi })(t,\cdot )\;\textrm{d}\varvec{x}\right] ^{t = \tau }_{t = 0} + \int _{0}^{\tau }\int _{{{\Omega }}}\big [\mu \nabla _{h}\varvec{u}_h:\nabla _{\varvec{x}}\varvec{\varphi }+\nu \,\textrm{div}_{{{h}}}(\varvec{u}_h)\,\textrm{div}_{\varvec{x}}(\varvec{\varphi })\big ]\,\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^\beta ) \nonumber \\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}\big [\varrho _h\overline{\varvec{u}_h}\cdot \partial _t\varvec{\varphi } +\varrho _h\overline{\varvec{u}_h}\otimes \overline{\varvec{u}_h}:\nabla _{\varvec{x}}\varvec{\varphi } +\left( p(\varrho _h\theta _{h})+h^\delta \big [\varrho _h^2+(\varrho _h\theta _{h})^2\big ] \right) \textrm{div}_{\varvec{x}}(\varvec{\varphi })\big ]\,\textrm{d}\varvec{x}\textrm{d}t \end{aligned}$$

for all \(\varvec{\varphi }\in C^{1}(\overline{\Omega _T})^d\), \(\varvec{\varphi }|_{[0,T]\times \partial \Omega }=\varvec{0}\), as \(h\downarrow 0\),

$$\begin{aligned} \left[ \,\int _{{{\Omega }}}(\varrho _h\theta _{h}\varphi )(t,\cdot )\;\textrm{d}\varvec{x}\right] ^{t = \tau }_{t = 0} = \int _{0}^{\tau }\int _{{{\Omega }}}\big [\varrho _h\theta _{h}\partial _t\varphi +\varrho _h\theta _{h}\overline{\varvec{u}_h} \cdot \nabla _{\varvec{x}}\varphi \big ]\,\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^\beta ) \end{aligned}$$

for all \(\varphi \in C^{1}(\overline{\Omega _T})\) as \(h\downarrow 0\) and

$$\begin{aligned} \left[ \,\int _{{{\Omega }}}(\varrho _h\ln (\theta _{h})\psi )(t,\cdot )\;\textrm{d}\varvec{x}\right] ^{t = \tau }_{t = 0} \ge \int _{0}^{\tau }\int _{{{\Omega }}}\big [\varrho _h\ln (\theta _{h})\partial _t\psi +\varrho _h\ln (\theta _{h})\,\overline{\varvec{u}_h}\cdot \nabla _{\varvec{x}}\psi \big ]\,\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^\beta ) \end{aligned}$$

for all \(\psi \in C^{1}(\overline{\Omega _T})\), \(\psi \ge 0\), as \(h\downarrow 0\). Here, the constants in the \({\mathcal {O}}\) notation do not depend on the particular time \(\tau \in [0,T]\).


The proof is given in Appendix A.3. \(\square \)

4.5 Error Estimates

We continue with the derivation of a priori error estimates for the FE-FV method. For convenience, we agree that in this section the constants hidden in the \(\lesssim \) -symbols and the \({\mathcal {O}}\) notation neither depend on the times \(\tau \in [0,T]\) nor on the number \(\alpha >0\) that will appear in the sequel.

4.5.1 Discrete Relative Energy

To begin with, we introduce a suitable extension of the relative energy \(E(\,\cdot \,|\,\cdot \,)\) that we will refer to as the discrete relative energy. It will be used to measure a “distance” between a numerical solution \((\varrho _h,\theta _{h},\varvec{u}_h)\) and a triplet \((\varrho ,\theta ,\varvec{u})\) of functions of the class (4.1) and reads

Our aim is to repeat the proof of Lemma 3.2 on the numerical level to obtain a version of the relative energy inequality for . First, our initial observation is that


for every \(\tau \in [0,T]\). To be able to transfer the next step in the proof of Lemma 3.2 to the discrete setting, we need to derive a suitable analogue of (3.8).

4.5.2 Partial Integration for Diffusion Terms

For the treatment of the diffusion terms, we extend the velocity to a function using Stein’s extension operator \({\mathfrak {E}}_{\textrm{Stein}}\), see [20, Chapter VI, Theorem 5], i.e., we put \({\widehat{\varvec{u}}}(t,\cdot )={\mathfrak {E}}_{\textrm{Stein}}[\varvec{u}(t,\cdot )]\)Footnote 6. As a consequence,


where \(C_\textrm{Stein}(\Omega ,2)>0\) is given by

Having extended \(\varvec{u}\) as described above, we may use Gauss’s theorem to observe that

$$\begin{aligned}&-\int _{0}^{\tau }\int _{\Omega _h}\nu \,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\,\textrm{div}_{{{h}}}(\varvec{u}_h-{\widehat{\varvec{u}}})\;\textrm{d}\varvec{x}\textrm{d}t\\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}\nu \,|\textrm{div}_{\varvec{x}}(\varvec{u})|^2\;\textrm{d}\varvec{x}\textrm{d}t- \int _{0}^{\tau }\int _{\Omega _h}\nu \,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}}) \,\textrm{div}_{{{h}}}(\varvec{u}_h)\;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h) \\&\quad = -\int _{0}^{\tau }\int _{{{\Omega }}}\nu \,\nabla _{\varvec{x}}\textrm{div}_{\varvec{x}}(\varvec{u})\cdot \varvec{u}\;\textrm{d}\varvec{x}\textrm{d}t-\int _{0}^{\tau }\sum _{K \,\in \, {\mathcal {T}}_h}\int _{{{K}}}\nu \,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\,\textrm{div}_{\varvec{x}}(\varvec{u}_h)\;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h) \\&\quad = -\int _{0}^{\tau }\int _{{{\Omega }}}\nu \,\nabla _{\varvec{x}}\textrm{div}_{\varvec{x}}(\varvec{u})\cdot \varvec{u}\;\textrm{d}\varvec{x}\textrm{d}t+ \int _{0}^{\tau }\sum _{K \,\in \, {\mathcal {T}}_h}\int _{{{K}}}\nu \,\nabla _{\varvec{x}}\,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\cdot \varvec{u}_h\;\textrm{d}\varvec{x}\textrm{d}t\\&\qquad - \int _{0}^{\tau }\int _{{\mathcal {E}}(K)}\nu \,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\,\varvec{u}_h\cdot \varvec{n}_K \;\textrm{d}S_{\varvec{x}}\textrm{d}t+ {\mathcal {O}}(h) \\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}\nu \,\nabla _{\varvec{x}}\textrm{div}_{\varvec{x}}(\varvec{u})\cdot (\varvec{u}_h-\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ \int _{0}^{\tau }\int _{{\mathcal {E}}}\nu \,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\,\llbracket \varvec{u}_h\rrbracket \cdot \varvec{n}_\sigma \;\textrm{d}S_{\varvec{x}}\textrm{d}t+ {\mathcal {O}}(h^{1/2}) \\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}\nu \,\nabla _{\varvec{x}}\textrm{div}_{\varvec{x}}(\varvec{u})\cdot (\varvec{u}_h-\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^{1/2}) \end{aligned}$$

for all \(\tau \in [0,T]\). Here, the first, the fourth and the fifth equality are due to (4.4), (4.31) and the first and the last estimate in (4.14) which yield


Similarly, we deduce that

$$\begin{aligned} -\int _{0}^{\tau }\int _{\Omega _h}\mu \nabla _{\varvec{x}}{\widehat{\varvec{u}}}:\nabla _{h}(\varvec{u}_h-{\widehat{\varvec{u}}})\;\textrm{d}\varvec{x}\textrm{d}t= \int _{0}^{\tau }\int _{{{\Omega }}}\mu \,\Delta _{\varvec{x}}\varvec{u}\cdot (\varvec{u}_h-\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^{1/2}) \end{aligned}$$

for all \(\tau \in [0,T]\). In addition, we may employ (4.4), (4.31) and the first and second estimate in (4.14) to observe that

Combining the previous estimates, we obtain the subsequent analogue of (3.8):

$$\begin{aligned}&-\int _{0}^{\tau }\int _{\Omega _h}\big [\nu \,\textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\,\textrm{div}_{{{h}}}(\varvec{u}_h-{\widehat{\varvec{u}}})+\mu \nabla _{\varvec{x}}{\widehat{\varvec{u}}}: \nabla _{h}(\varvec{u}_h-{\widehat{\varvec{u}}})\big ]\,\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&-\int _{0}^{\tau }\int _{\Omega _h\backslash \Omega }\big [\mu \nabla _{h}\varvec{u}_h:\nabla _{\varvec{x}}{\widehat{\varvec{u}}}+\nu \,\textrm{div}_{{{h}}}(\varvec{u}_h)\, \textrm{div}_{\varvec{x}}({\widehat{\varvec{u}}})\big ]\,\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}\textrm{div}_{\varvec{x}}({\mathbb {S}}(\nabla _{\varvec{x}}\varvec{u}))\cdot (\varvec{u}_h-\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^{1/2}) \nonumber \\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}\textrm{div}_{\varvec{x}}({\mathbb {S}}(\nabla _{\varvec{x}}\varvec{u})) \cdot (\overline{\varvec{u}_h}-\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^{1/2}) \nonumber \\&\quad = \int _{0}^{\tau }\int _{{{\Omega }}}(\varrho _h-\varrho )\,\frac{1}{\varrho }\,\textrm{div}_{\varvec{x}}({\mathbb {S}} (\nabla _{\varvec{x}}\varvec{u}))\cdot (\varvec{u}-\overline{\varvec{u}_h}) -\frac{\varrho _h}{\varrho }\,(\varvec{u}-\overline{\varvec{u}_h})\cdot \textrm{div}_{\varvec{x}}({\mathbb {S}}(\nabla _{\varvec{x}}\varvec{u})) \;\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^{1/2}) \end{aligned}$$

for all \(\tau \in [0,T]\), where the second equality is due to the first estimate in (A.3) and the first estimate in (4.14) which imply

4.5.3 Relative Energy Inequality for —General Form

It is now easy to transfer the remaining part of the proof of Lemma 3.2 to the discrete setting. Indeed, starting from (4.30) and ignoring the \(h^\delta \)-terms, a repetition of the steps of the proof of Lemma 3.2 using (4.24), (4.26)–(4.29) and (4.32) instead of (2.3), (2.4)–(2.7) and (3.8) yields

for all \(\tau \in [0,T]\). Then, using (4.26) and (4.28), we easily verify that

$$\begin{aligned}&- h^\delta \int _{0}^{\tau }\int _{{{\Omega }}}\varrho _h^2\big [1+\theta _{h}^2\big ]\,\textrm{div}_{\varvec{x}}(\varvec{u})\;\textrm{d}\varvec{x}\textrm{d}t+ \left[ h^\delta \int _{{{\Omega }}}\varrho ^{\,2}(t,\cdot )\left[ 1+\theta ^{\,2}(t,\cdot )\right] \textrm{d}\varvec{x}\right] _{t = 0}^{t = \tau } \\&- 2h^\delta \left[ \,\int _{{{\Omega }}}\varrho _h(t,\cdot )\varrho (t,\cdot )\;\textrm{d}\varvec{x}\right] _{t = 0}^{t = \tau } - 2h^\delta \left[ \,\int _{{{\Omega }}}\varrho _h(t,\cdot )\theta _{h}(t,\cdot )\varrho (t,\cdot ) \theta (t,\cdot )\;\textrm{d}\varvec{x}\right] _{t = 0}^{t = \tau } \\&\quad = 2h^\delta \int _{0}^{\tau }\int _{{{\Omega }}}\varrho _h(\varvec{u}-\overline{\varvec{u}_h})\cdot \nabla _{\varvec{x}}\varrho \;\textrm{d}\varvec{x}\textrm{d}t+ 2h^\delta \int _{0}^{\tau }\int _{{{\Omega }}}\varrho _h\theta _{h}(\varvec{u}-\overline{\varvec{u}_h})\cdot \nabla _{\varvec{x}}(\varrho \theta )\;\textrm{d}\varvec{x}\textrm{d}t\\&\quad + 2h^\delta \int _{0}^{\tau }\int _{{{\Omega }}}(\varrho -\varrho _h)\,\big [\partial _t\varrho +\textrm{div}_{\varvec{x}}(\varrho \varvec{u}) \big ]\,\textrm{d}\varvec{x}\textrm{d}t\\&\quad + 2h^\delta \int _{0}^{\tau }\int _{{{\Omega }}}(\varrho \theta -\varrho _h\theta _h)\,\big [\partial _t(\varrho \theta )+\textrm{div}_{\varvec{x}}(\varrho \theta \varvec{u})\big ]\,\textrm{d}\varvec{x}\textrm{d}t+ {\mathcal {O}}(h^\beta ) \end{aligned}$$

for all \(\tau \in [0,T]\). Consequently,


for all \(\tau \in [0,T]\).

4.5.4 Relative Energy Inequality for —Reduced form for Strong Solutions (Part I)

In a particular situation when \((\varrho ,\theta ,\varvec{u})\) is a strong solution to (1.1)–(1.5) of the class (4.1), the relative energy inequality (4.33) reduces to


Our goal is now to rewrite (4.34) in such a way that we can apply Gronwall’s lemma. To this end, we first consider the terms \(T_j\), \(j\in \{1,\dots ,8\}\). Clearly,

Moreover, the second and third estimate in (4.13) yield

Then, exactly as in the proof of Theorem 3.3, we see that

To handle the term \(T_8\), we need a suitable analogue of (2.8).

4.5.5 A Discrete Analogue of Poincaré’s Inequality (2.8)

For the derivation of the discrete analogue of (2.8) it shall be convenient to introduce the following notation:

$$\begin{aligned} {\mathcal {T}}_{h,\,\textrm{ext}}=\{K\in {\mathcal {T}}_h\,|\,{\mathcal {E} }_\textrm{ext}(K)\ne \emptyset \} \qquad \text {and} \qquad \Omega _{h,\,\textrm{ext}} = \textrm{int}\left[ \bigcup _{K\,\in \,{\mathcal {T}}_{h,\,\textrm{ext}}} K\right] . \end{aligned}$$

With this notation at hand, we observe that

$$\begin{aligned}&\int _{0}^{\tau }\int _{\Omega _h}|\overline{\varvec{u}_h}-{\widehat{\varvec{u}}}|^2\;\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&\quad \lesssim \int _{0}^{\tau }\int _{\Omega _h}\left( |\overline{\varvec{u}_h}-\varvec{u}_h|^2 + |\varvec{u}_h-\Pi _{V,h}^0{\widehat{\varvec{u}}}|^2 + |\Pi _{V,h}^0{\widehat{\varvec{u}}}-\Pi _{V,h}{\widehat{\varvec{u}}}|^2 + |\Pi _{V,h}{\widehat{\varvec{u}}}-{\widehat{\varvec{u}}}|^2\right) \,\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&\quad \lesssim h^2 + \int _{0}^{\tau }\int _{\Omega _h}|\nabla _{h}(\varvec{u}_h-\Pi _{V,h}^0{\widehat{\varvec{u}}})|^2\;\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&\quad \lesssim h^2 + \int _{0}^{\tau }\int _{\Omega _h}\left( |\nabla _{h}(\varvec{u}_h-{\widehat{\varvec{u}}})|^2 + |\nabla _{h}({\widehat{\varvec{u}}}-\Pi _{V,h}{\widehat{\varvec{u}}})|^2 + |\nabla _{h}(\Pi _{V,h}{\widehat{\varvec{u}}}-\Pi _{V,h}^0{\widehat{\varvec{u}}})|^2\right) \,\textrm{d}\varvec{x}\textrm{d}t\nonumber \\&\quad \lesssim h + \int _{0}^{\tau }\int _{\Omega _h}|\nabla _{h}(\varvec{u}_h-{\widehat{\varvec{u}}})|^2\;\textrm{d}\varvec{x}\textrm{d}t\,, \end{aligned}$$

where in the second step we have used (A.11) as well as the estimates

which are based on the first estimate in (A.3) and the estimates (4.31), (A.13), (A.15). The last step in (4.35) is due to the estimates

that are based on (A.14), (4.31), (A.15).

4.5.6 Relative Energy Inequality for —Reduced form for Strong Solutions (Part II)

With the help of (4.35) we can now estimate the term \(T_8\) in (4.34) analogously to its continuous counterpart. We obtain

for all \(\alpha >0\). Together with the estimates for the terms \(T_j\), \(j\in \{1,\dots ,7\}\), stated in Sect. 4.5.4, this observation allows us to rewrite (4.34) as


for all \(\alpha >0\). Next, let us turn to the first line in (4.36). Using Hölder’s inequality, the first estimate in (A.3), (A.13) and (A.16), we deduce that

Moreover, denoting

$$\begin{aligned} {\mathcal {T}}_{h,\Omega }= & {} \{K\in {\mathcal {T}}\,|\,K\subset \Omega \}\,, \qquad \Omega _{h,\Omega }=\textrm{int}\left[ \bigcup _{K\,\in \,{\mathcal {T} }_{h,\Omega }}K\right] ,\\ A(\varrho _0,S_0)= & {} [(\varrho _0)_\star ,(\varrho _0)^\star ]\times \big [-(\varrho _0 )^\star \max \{|\ln ((\theta _{0})_\star )|,|\ln ((\theta _{0})^\star )|\},(\varrho _0)^\star \max \{|\ln ((\theta _{0})_\star )|,| \ln ((\theta _{0})^\star )|\}\big ]\,, \end{aligned}$$

and employing Hölder’s inequality, Taylor’s theorem, (A.12) and (A.16), we observe that

Consequently, we may rewrite (4.36) as


Fixing a sufficiently small \(\alpha >0\), we deduce from (4.37) the inequality


4.5.7 Error Estimates

We are now ready to apply Gronwall’s lemma to (4.38) which yields


Combining (4.39) with (4.35), we get

Furthermore, using Lemma A.1 and \(\theta _\star \le \theta _{h}\le \theta ^\star \), it is easy to see that for all \(p\in [1,\gamma ]\), all \(q\in [1,\infty )\) and all \(\tau \in [0,T]\)

Consequently, (4.40) yields

Remark 4.4

The optimal convergence rates are obtained for \(\varepsilon \ge 7/6\) and \(\delta =1/6\). In this case, \(\min \{\beta ,\delta \}=1/6\) and, in particular, the convergence rates for \(\varvec{u}_h\) in the -norm, for \(\varrho _h\) in the --norm (provided \(\gamma \le 2\)) and for \(\theta _{h}\) in the -norm are 1/12.

4.6 Numerical Results

We conclude this section by illustrating experimentally convergence behaviour of the FE-FV method (4.5)–(4.7). More specifically, motivated by the numerical experiments presented in [21, Section 5.1] and [7, Chapter 14.6.2], we simulate a vortex flow in \(\Omega =[0,1]^2\subset {\mathbb {R}}^2\) with the initial data

$$\begin{aligned} \varrho _0 \equiv 1\,, \qquad \theta _{0}(\varvec{x}) = \frac{1}{2}+\frac{1}{4}\,\theta _r(\varvec{x})\,, \qquad \varvec{u}_0(\varvec{x}) = u_r(\varvec{x})\left( \begin{array}{c} x_2-\frac{1}{2} \\ \frac{1}{2}-x_1 \end{array}\right) , \end{aligned}$$


$$\begin{aligned} r(\varvec{x})= & {} \sqrt{\left( x_2-\frac{1}{2}\right) ^{2}+\left( \frac{1}{2}-x_1\right) ^{2}}\,, \qquad u_r(\varvec{x})=\sqrt{\gamma }\left\{ \begin{array}{ll} 10 &{}\quad \text {if }r(\varvec{x})<\frac{1}{10}, \\ 2\left( \frac{1}{r(\varvec{x})}-5\right) &{}\quad \text {if }\frac{1}{10}\le r(\varvec{x})<\frac{1}{5}, \\ 0 &{}\quad \text {if }r(\varvec{x})\ge \frac{1}{5}, \end{array}\right. \\ \theta _r(\varvec{x})= & {} \left\{ \begin{array}{ll} 50r(\varvec{x})^2 &{} \quad \text {if }r(\varvec{x})<\frac{1}{10}, \\ 4\ln (10r(\varvec{x}))+4-40r(\varvec{x})+50r(\varvec{x})^2 &{}\quad \text {if }\frac{1}{10}\le r(\varvec{x})<\frac{1}{5},\\ 4\ln (2)-2 &{}\quad \text {if }r(\varvec{x})\ge \frac{1}{5}. \end{array}\right. \end{aligned}$$

The parameters of the FE-FV method are chosen as \(\mu =0.1\), \(\nu =0\), \(\varepsilon =2.0\), \(\delta =0.1667\) and the final time for our convergence study is \(T=0.1\). The nonlinear algebraic system (4.5)–(4.7) is solved using a fixed point iteration. Thus, in each subiteration, the CFL stability condition

is required. This is ensured by the choice \(\Delta t=16h/130\). We concentrate on the following errors:

where \(h_\textrm{ref}=1/1024\) and

$$\begin{aligned} E(\varrho _h,\theta _{h},\varvec{u}_h|\varrho _{h_\textrm{ref}}, \theta _{h_\textrm{ref}},\varvec{u}_{h_\textrm{ref}})&= \frac{1}{2}\,\varrho _h|\overline{\varvec{u}_h}-\overline{\varvec{u}_{h_\textrm{ref}}}|^2 \\&\quad + P(\varrho _h,S_h) -\frac{\partial P(\varrho _{h_\textrm{ref}}, S_{h_\textrm{ref}})}{\partial \varrho }\,(\varrho _h-\varrho _{h_\textrm{ref}})\\&\quad -\frac{\partial P(\varrho _{h_\textrm{ref}},S_{h_\textrm{ref}})}{\partial S} \,(S_h-S_{h_\textrm{ref}}) - P(\varrho _{h_\textrm{ref}},S_{h_\textrm{ref}})\,. \end{aligned}$$

Tables 1 and 2 show the experimental order of convergence for two different values of the adiabatic exponent \(\gamma = 1.4\) and \(\gamma = 1.67.\)

Table 1 Convergence study for the FE-FV method with \(\gamma =1.4\)
Table 2 convergence study for the FE-FV method with \(\gamma =1.67\)

Here, the experimental orders of convergence were computed using the standard formula

$$\begin{aligned} \textrm{EOC}(h)=\log _2\left( \frac{||\varvec{s}_{2h}-\varvec{s}_{h_\textrm{ref}}||}{||\varvec{s}_{h}-\varvec{s}_{h_\textrm{ref}}||}\right) , \end{aligned}$$

where \(\varvec{s}_h\) stands for a numerical solution on a mesh \(\Omega _h,\) analogous notations are used for \(\varvec{s}_{2h}\) and \(\varvec{s}_{h_\textrm{ref}}.\) We observe that EOC for the density, velocity, gradient of velocity and potential temperature are around 1, while the second order EOC are obtained for the relative energy. Similarly as in theoretical analysis the convergence rates in the relative energy are twice as good as those of the density, velocity and potential temperature. Our numerical experiments indicate that theoretical results obtained in Sect. 4.5 might be suboptimal, such a behaviour was observed in the literature also for other numerical methods and models, see, e.g., [9, 13, 16]. Figure 1 illustrates time evolution of the solution computed at different times on a mesh with \(h=1/128\) and for \(\gamma =1.4.\)

Fig. 1
figure 1

Numerical solutions for \(\varrho \), \(\varrho \theta \), \(u_1\), \(u_2\) at times \(t=0,0.1,0.2,0.5\)

5 Conclusions

In the present paper, we have proved the DMV-strong uniqueness principle for the Navier–Stokes system with potential temperature transport (1.1)–(1.5). This result shows that strong solutions are stable in the class of DMV solutions introduced in [15]. We have derived the relative energy by taking the total physical entropy into account. More precisely, the pressure was rewritten as a function of the density and entropy, instead of the total potential temperature only. Moreover, we also require the entropy inequality (2.7) that is included in our definition of DMV solutions. The importance of Poincaré’s inequality (2.8) became clear from the proof of DMV-strong uniqueness: It allowed us to rewrite viscosity terms in such a way that Gronwall’s lemma was applicable and yield the DMV-strong uniqueness principle.

As an application of the DMV-strong uniqueness principle we derive a priori error estimates by applying the relative energy to numerical solutions. Our theoretical error estimates include not only the errors between the numerical and the strong solutions but also the so-called variational crime errors due to the approximation of a smooth domain \(\Omega \) by polygonal approximations \(\Omega _h\), \(\Omega \subset \Omega _h\) such that \(\textrm{dist}(\varvec{x},\partial \Omega )={\mathcal {O}}(h)\) for all \(\varvec{x}\in \partial \Omega _h.\)