Compressible Navier–Stokes Equations with Potential Temperature Transport: Stability of the Strong Solution and Numerical Error Estimates

We present a dissipative measure-valued (DMV)-strong uniqueness result for the compressible Navier–Stokes system with potential temperature transport. We show that strong solutions are stable in the class of DMV solutions. More precisely, we prove that a DMV solution coincides with a strong solution emanating from the same initial data as long as the strong solution exists. As an application of the DMV-strong uniqueness principle we derive a priori error estimates for a mixed finite element-finite volume method. The numerical solutions are computed on polyhedral domains that approximate a sufficiently a smooth bounded domain, where the exact solution exists.

The relative energy inequality corresponding to (3.4) reads as follows.
for a.a. τ ∈ (0, T ). Here, denotes the absolute temperature. Proof. 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 . (3.9) 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 ∈ {t, x 1 , . . . , x d }, 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.
From the relative energy inequality we can deduce the DMV-strong uniqueness result.
and the DMV and strong solutions coincide on [0, T * ), i.e.
Remark 3.4. The local existence of strong solutions to (1.1)-(1.5) for the Cauchy problem (i.e. Ω = 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 Ω is of class C 3 and the initial data satisfy suitable compatibility conditions. This can be an interesting topic for future studies.

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 Ω by a sequence of polygonal computational domains Ω h , h ↓ 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.

Regularity Class for the Strong Solution and the Initial Data
We will consider strong solutions ( , θ, u) to (1.1)-(1.5) that belong to the regularity class Accordingly, the initial data satisfy For functions such as in (4.1) and 0 in (4.2), we further introduce the following notation: In addition, we consider the initial data ( 0 , θ 0 , u 0 ) to be extended by (( 0 ) , (θ 0 ) , 0) outside Ω.

Mixed Finite Element-Finite Volume Method
We recall the mixed FE-FV numerical method introduced in [ We assume that the following conditions are satisfied: • There is a constant D > 0 such that • Each element K of a mesh T h is a d-simplex that can be written as The symbol E h denotes the set of all faces, d = 3, or all edges, d = 2, in the mesh T h . E h,ext and E h,int stand for the sets of exterior and interior faces, respectively, i.e., In connection with these sets, we shall use the abbreviations Each face σ ∈ E h is equipped with a unit vector n σ that is determined as follows: We fix an arbitrary element K σ ∈ T h such that σ ∈ E h (K σ ) and set n σ = n Kσ (x σ ). Here, x σ denotes the center of mass of σ and n Kσ (x σ ) is the outward-pointing unit normal vector to the element K σ at x σ . Finally, it will be convenient to write A B whenever there is an h-independent constant c > 0 such that A ≤ cB and A ≈ B whenever A B and B A.
The Crouzeix-Raviart finite element spaces are denoted by With these spaces we associate the projections Π V,h : respectively. Additionally, we agree on the notation

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 ∇ x and div x . They are determined by the stipulations respectively. We continue by recalling the trace operators. For arbitrary The convective terms shall be approximated by means of a dissipative upwind operator.
where Δt > 0 is a given time step and s k−1 h and s k h are the numerical solutions at the time levels t k−1 = (k − 1)Δt and t k = kΔt, respectively. For the sake of simplicity, we assume that Δt is constant and that there is a number N T ∈ N such that N T Δt = T .

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

Discrete Initial Data.
The initial data for the mixed FE-FV method (4.5)-(4.7) are determined as follows: As a consequence of this stipulation, we observe that ( 0

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 ∈ N 0 . (4.10) In addition, it turns out that the numerical solutions satisfy an energy balance and an entropy inequality that read as follows:

Discrete entropy inequality: (cf. [15, Lemma 4.5])
For every k ∈ N and every pair (χ, In addition, we introduce the functions Next, let us state two consequences of the discrete energy balance (4.11).

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 Ω (we now have Ω ⊂ Ω h instead of Ω h ⊂ Ω) 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 Energy inequality:

Consistency
We proceed by stating a suitable consistency formulation of the numerical scheme (4.5)-(4.7).
Here, the constants in the O notation do not depend on the particular Proof. The proof is given in Appendix A.3.

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 -symbols and the O notation neither depend on the times τ ∈ [0, T ] nor on the number α > 0 that will appear in the sequel.

Discrete Relative Energy.
To begin with, we introduce a suitable extension of the relative energy E( · | · ) that we will refer to as the discrete relative energy. It will be used to measure a "distance" between a numerical solution ( h , θ h , u h ) and a triplet ( , θ, 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 forẼ( ·| · ). First, our initial observation is that for every τ ∈ [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).   6 . As a consequence, where C Stein (Ω, 2) > 0 is given by .
Having extended u as described above, we may use Gauss's theorem to observe that 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 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): for all τ ∈ [0, T ], where the second equality is due to the first estimate in (A.3) and the first estimate in (4.14) which imply h .

Relative Energy Inequality forẼ-Reduced form for Strong Solutions (Part I).
In a particular situation when ( , θ, 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 ∈ {1, . . . , 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).

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: With this notation at hand, we observe that 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 h that are based on (A.14), (4.31), (A.15).

Relative Energy Inequality forẼ-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 α > 0. Together with the estimates for the terms T j , j ∈ {1, . . . , 7}, stated in Sect. 4.5.4, this observation allows us to rewrite (4.34) as Moreover, denoting and employing Hölder's inequality, Taylor's theorem, (A.12) and (A.16), we observe that h .

Numerical Results
We conclude this section by illustrating experimentally convergence behaviour of the FE-FV method where The parameters of the FE-FV method are chosen as μ = 0.1, ν = 0, ε = 2.0, δ = 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 Δt = 16h/130. We concentrate on the following errors: , , , , . Tables 1 and 2 show the experimental order of convergence for two different values of the adiabatic exponent γ = 1.4 and γ = 1.67.
Here, the experimental orders of convergence were computed using the standard formula where s h stands for a numerical solution on a mesh Ω h , analogous notations are used for s 2h and s h 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 γ = 1.4.

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 Ω by polygonal approximations Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A.1. An Auxiliary Result Concerning the Relative Energy
Here, we prove the auxiliary result used in the proof of DMV-strong uniqueness.
Proof. To begin with, let 0 < c 1 ≤ c 2 , and c 3 ≥ c / θ be arbitrary numbers. Further, let R, S be defined as described in the lemma. We decompose S into the sets and observe that Here, the first inequality is obtained using Young's inequality. Together, the above observations show that we can specify c 1 , c 2 , c 3 in dependence of , , θ, θ, c , γ such that where c 4,1 > 0 solely depends on , , θ, θ, c , γ. Having fixed c 1 , c 2 , c 3 as described above, it remains to show that where c 4,2 > 0 only depends on , , θ, θ, c , γ. This inequality is a direct consequence of the fact that P = P ( , S) is strongly convex on every compact convex subset of (0, ∞) × R which, in turn, follows from the positive definiteness of the Hessian of P on (0, ∞) × R.

A.2. Mesh-Related Estimates
We recall several important mesh-related estimates; see, e.g., [7] and the references therein. We begin with the discrete trace and inverse inequalities. We have and ||r|| are valid for all p ∈ [1, ∞], all v ∈ V 0,h , all K ∈ T h , and all σ ∈ E h (K). Moreover, for all K ∈ T h and all σ ∈ E h (K), (A.7) .
for all q ∈ [1, ∞], all φ ∈ W 1,q (Ω h ), and all ψ ∈ W 2,q (Ω h ). Furthermore, we record the following estimate concerning the comparison of the operators Π V,h and Π 0 V,h , see [4,Corollary 2.12]: Finally, we report the boundedness of the projection operators Π Q,h and Π V,h . It follows from Jensen's inequality, cf. [5, p.90], that Furthermore, we may use (A.13) and the triangle inequality to deduce that there exists an h-independent constant C > 0 such that

A.3. Proof of Theorem 4.3
This appendix is devoted to the proof of Theorem 4.3. Its proof is an adaption of the proof of [15,Theorem 5.1]. Apart from the estimates listed in Appendix A.2, we need the subsequent results.
Together, the previous computations yield the desired result.

The continuity equation.
From (4.5) we deduce that Using the fact that h (t, ·) ∈ Q h for every t ∈ [0, τ], we see that Consequently, (4.4), the second estimate in (4.15), (A.16) and (A.21) yield Next, let us consider the second term on the left-hand side of (A.22). Employing Hölder's inequality, the first estimate in (A.3), the second estimate in (4.15), the first estimate in (4.14) and (A.21), we see that Moreover, applying (A.21), (4.4), the second estimate in (4.15) and the first estimate in (4.13), we obtain These terms can be further estimated as follows.
• Term |I 2,h |. Due to (4.21), we obtain • Term |I 3,h |. By means of Hölder's inequality, the second estimate in (A.3), the first estimate in (A.2), the second estimate in (4.15), and the first estimate in (4.14), we derive • Term |I 4,h |. Employing Hölder's inequality, the second estimate in (4.14), and the second estimate in (4.15), we conclude that • Term |I 5,h |. Applying the first estimate in (A.2) and the second estimate in (4.13), we get Consequently, where α 1 = min ε, 1−δ 4 . The potential temperature equation. The proof of (4.28) can be done by repeating the proof of (4.26) with h and 0 h replaced by h θ h and 0 h θ 0 h , respectively.

The momentum equation.
Realizing that ϕ h (t, ·) ∈ V 0,h for all t ∈ [0, T ], we deduce from (4.7) that Let us consider the first term on the left-hand side of (A.25). Since ϕ vanishes on where by Hölder's inequality, the second information in (4.10), (A.16), the second estimate in (4.19), (A.13) and Δt ≈ h and by Hölder's inequality, the first estimate in (4.23), the third estimate in (4.14), (A.16), (A.5) and In view of the previous two computations, it is easy to verify that as h ↓ 0. Next, we turn to the last three terms on the left-hand side of (A.25). It follows from Lemma A.5 that Finally, let us examine the second term on the left-hand side of (A.25). Using Hölder's inequality, the first estimate in (A.3), the second estimate in (4.16) and the first estimate in (4.14), we deduce that We continue by estimating the above terms.

The entropy inequality.
Taking ψ h (t, ·) = ψ h (t, ·) + C Stein (Ω T , 1)Dh ||ψ|| . Now we may rewrite the first two integrals in (A.30) following the procedure used to handle the continuity equation. The error terms appearing during this process are exactly the same as in the case of the continuity equation with h replaced by h ln(θ h ) and ϕ h replaced by ψ h . However, the analogue of the error term I 5,h will not be there since (A.30) contains the usual upwind operator Up[ · , · ] instead of the dissipative upwind operator F up for every (k, σ) ∈ N × E int and suitably chosen values (η θ,k,σ ) σ ∈ Eint ⊂ [(θ 0 ) , (θ 0 ) ] and, analogously, it is easy handle these error terms. Moreover, using (θ 0 ) ≤ θ h ≤ (θ 0 ) , the second information in (4.10) and (A.16), we easily verify that h .
Proof of Theorem 4.3 (Part II). In this part, we turn to the situation in which τ ∈ [0, T ] is arbitrary. Since the case τ = 0 is trivial, we may assume without loss of generality that τ ∈ (0, T ]. Let m ∈ {1, . . . , N T } be the smallest number such that t m = mΔt ≥ τ .

The continuity equation.
Using Hölder's inequality, (A.16) and Δt ≈ h, we deduce that Ω ( h ϕ)(t, ·) dx t=tm t=τ Δt ||ϕ|| Moreover, employing Hölder's inequality the second and third estimate in (4.13), we see that The potential temperature equation and the entropy inequality. Keeping in mind that (θ 0 ) ≤ θ h ≤ (θ 0 ) , one easily reduces the setting of the potential temperature equation and the entropy inequality to that of the continuity equation.