Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability

Aaron Brunk; Oliver Habrich; Timileyin David Oyedeji; Yangyiwei Yang; Bai-Xiang Xu

doi:10.1515/cmam-2023-0274

Article

Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability

Aaron Brunk , Oliver Habrich , Timileyin David Oyedeji , Yangyiwei Yang and Bai-Xiang Xu

Published/Copyright: September 3, 2024

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From the journal Computational Methods in Applied Mathematics Volume 25 Issue 2

Abstract

A Cahn–Hilliard–Allen–Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

Keywords: Non-Isothermal Phase-Field; Cross-Diffusion; Finite Elements; Entropy Stable

MSC 2020: 65M60; 35K52; 35K55; 65M12; 82C26

Funding statement: Financial support by the German Science Foundation (DFG, Project number 441153493) within the Priority Program SPP 2256: Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials (project BR 7093/1-2 and Xu 121/13-2) and via TRR 146: Multiscale Simulation Methods for Soft Matter Systems (project C3) is gratefully acknowledged. Part of the research was conducted during a research stay of the first author at RICAM/JKU Linz.

A Proof Full Discrete Stability Estimate

In this section, we will expand the fully discrete relative entropy. We divide the temporal jump of the relative entropy as follows:

The structure of this appendix is as follows:

The first term from the above decomposition will be expanded in Section A.1
The resulting relative dissipation is estimated in Section A.3
In Section A.4 we deal with the quadratic terms from above.
Finally, in Section A.5 we collect all parts together

To increase readability, we will neglect the index h.

A.1 Expansion Relative Entropy

Let us compute the evolution of the discrete relative energy, adding suitable zeros and rearranging terms, we find

𝒲 ( ρ , θ , η | ρ ^ , θ ^ , η ^ ) | t n t n + 1 = 〈 ψ ( ρ n + 1 , θ n + 1 , η n + 1 ) - ψ ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) - ∂ θ ψ ( ρ n + 1 , θ n + 1 , η n + 1 ) ( θ n + 1 - θ ^ n + 1 )

- ∂ ρ ψ ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) ( ρ n + 1 - ρ ^ n + 1 ) - ∂ η ψ ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) ( η n + 1 - η ^ n + 1 ) , 1 〉

+ γ 2 ⁢ ∥ ∇ ⁡ ρ n + 1 ∥ 0 2 + γ 2 ⁢ ∥ ∇ ⁡ η n + 1 ∥ 0 2 - γ 2 ⁢ ∥ ∇ ⁡ ρ ^ n + 1 ∥ 0 2 - γ 2 ⁢ ∥ ∇ ⁡ η ^ n + 1 ∥ 0 2

- 〈 ψ ( ρ n , θ n , η n ) - ψ ( ρ ^ n , θ ^ n , η ^ n ) - ∂ θ ψ ( ρ n , θ n , η n ) ( θ n - θ ^ n )

- ∂ ρ ψ ( ρ ^ n , θ ^ n , η ^ n ) ( ρ n - ρ ^ n ) - ∂ η ψ ( ρ ^ n , θ ^ n , η ^ n ) ( η n - η ^ n ) , 1 〉

- γ 2 ⁢ ∥ ∇ ⁡ ρ n ∥ 0 2 - γ 2 ⁢ ∥ ∇ ⁡ η n ∥ 0 2 + γ 2 ⁢ ∥ ∇ ⁡ ρ ^ n ∥ 0 2 + γ 2 ⁢ ∥ ∇ ⁡ η ^ n ∥ 0 2

= - 〈 d n + 1 ⁢ ( e - e ^ ) , θ n + 1 - θ ^ n + 1 〉 + 〈 d n + 1 ⁢ ( ρ - ρ ^ ) , μ ρ n + 1 - μ ^ ρ n + 1 + r 2 n + 1 〉

+ 〈 d n + 1 ⁢ ( η - η ^ ) , μ η n + 1 - μ ^ η n + 1 + r 5 n + 1 〉

- 〈 e ⁢ ( ρ n , θ n , η n ) , d n + 1 ⁢ ( θ n + 1 - θ ^ n + 1 ) 〉 - 〈 d n + 1 ⁢ e ^ , θ n + 1 - θ ^ n + 1 〉 - γ 2 ⁢ ∥ ∇ ⁡ d n + 1 ⁢ ( ρ - ρ ^ ) ∥ 0 2

- γ 2 ⁢ ∥ ∇ ⁡ d n + 1 ⁢ ( η - η ^ ) ∥ 0 2 - 〈 d n + 1 ⁢ ( ρ - ρ ^ ) , ∂ ρ ⁡ ψ ~ - ∂ ρ ⁡ ψ ^ ~ 〉 - 〈 d n + 1 ⁢ ( η - η ^ ) , ∂ η ⁡ ψ ~ - ∂ η ⁡ ψ ^ ~ 〉

+ 〈 ψ ( ρ n + 1 , θ n + 1 , η n + 1 ) - ψ ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) - ∂ ρ ψ ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) ( ρ n + 1 - ρ ^ n + 1 )

- ∂ η ψ ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) ( η n + 1 - η ^ n + 1 ) , 1 〉

- 〈 ψ ( ρ n , θ n , η n ) - ψ ( ρ ^ n , θ ^ n , η ^ n ) - ∂ ρ ψ ( ρ ^ n , θ ^ n , η ^ n ) ( ρ n - ρ ^ n )

- ∂ η ψ ( ρ ^ n , θ ^ n , η ^ n ) ( η n - η ^ n ) , 1 〉

= - 〈 d n + 1 ⁢ ( e - e ^ ) , θ n + 1 - θ ^ n + 1 〉 + 〈 d n + 1 ⁢ ( ρ - ρ ^ ) , μ ρ n + 1 - μ ^ ρ n + 1 + r 2 n + 1 〉

+ 〈 d n + 1 ⁢ ( η - η ^ ) , μ η n + 1 - μ ^ η n + 1 + r 5 n + 1 〉 + ℛ .

A.2 Remainder ℛ

In order to rewrite and estimate the remainder term ℛ , we introduce the following notation:

ψ ( ρ n + 1 , θ n + 1 , η n + 1 ) = : ψ n + 1 ,

ψ ( ρ n , θ n , η n ) = : ψ n ,

ψ ( ρ n , θ n + 1 , η n ) = : ψ n , n + 1 , n .

In this new notation, the remainder term can be written as

ℛ = 〈 ψ n + 1 - ψ n - ∂ θ ⁡ ψ n ⁢ d n + 1 ⁢ θ - ∂ ρ ⁡ ψ vex n + 1 ⁢ d n + 1 ⁢ ρ - ∂ ρ ⁡ ψ cav n , n + 1 , n ⁢ d n + 1 ⁢ ρ - ∂ η ⁡ ψ vex n + 1 ⁢ d n + 1 ⁢ η - ∂ η ⁡ ψ cav n , n + 1 , n ⁢ d n + 1 ⁢ η , 1 〉

+ 〈 ψ ^ n + 1 - ψ ^ n - ∂ θ ⁡ ψ n ⁢ d n + 1 ⁢ θ ^ n + 1 - ∂ ρ ⁡ ψ ^ vex n + 1 ⁢ d ⁢ ρ ^ - ∂ ρ ⁡ ψ ^ cav n , n + 1 , n ⁢ d n + 1 ⁢ ρ ^ - ∂ η ⁡ ψ ^ vex n + 1 ⁢ d n + 1 ⁢ η ^ - ∂ η ⁡ ψ ^ cav n , n + 1 , n ⁢ d n + 1 ⁢ η ^ , 1 〉

+ 〈 ∂ θ ⁡ ψ n - ∂ θ ⁡ ψ ^ n , d n + 1 ⁢ θ ^ 〉 - 〈 ∂ θ ⁡ ψ ^ n + 1 - ∂ θ ⁡ ψ ^ n , ( θ n + 1 - θ ^ n + 1 ) 〉

- 〈 ∂ ρ ⁡ ψ ^ n + 1 , ρ n + 1 - ρ ^ n + 1 〉 + 〈 ∂ ρ ⁡ ψ ^ n , ρ n - ρ ^ n 〉 - 〈 ∂ η ⁡ ψ ^ n + 1 , η n + 1 - η ^ n + 1 〉 + 〈 ∂ η ⁡ ψ ^ n , η n - η ^ n 〉

+ 〈 ∂ ρ ⁡ ψ ^ vex n + 1 + ∂ ρ ⁡ ψ ^ cav n , n + 1 , n , d n + 1 ⁢ ρ 〉 + 〈 ∂ η ⁡ ψ ^ vex n + 1 + ∂ η ⁡ ψ ^ cav n , n + 1 , n , d n + 1 ⁢ η 〉

+ 〈 ∂ ρ ⁡ ψ vex n + 1 + ∂ ρ ⁡ ψ cav n , n + 1 , n - 2 ⁢ ∂ ρ ⁡ ψ ^ vex n + 1 + 2 ⁢ ∂ ρ ⁡ ψ ^ cav n , n + 1 , n , ρ ^ n + 1 - ρ ^ n 〉

+ 〈 ∂ η ⁡ ψ vex n + 1 + ∂ η ⁡ ψ cav n , n + 1 , n - 2 ⁢ ∂ η ⁡ ψ ^ vex n + 1 + 2 ⁢ ∂ η ⁡ ψ ^ cav n , n + 1 , n , η ^ n + 1 - η ^ n 〉

= (i) + … + (xii) ,

where we numbered the different inner products. For simplicity we introduce 𝐰 = ( ρ , η ) ⊤ . By adding and subtracting ψ n , n + 1 , n , ψ ^ n , n + 1 , n , we find that

2 ⁢ ( (i) + (ii) ) = 〈 ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , n ⁢ d n + 1 ⁢ θ , d n + 1 ⁢ θ 〉 + 〈 ( H 𝐰 cav , χ 1 , n + 1 , χ 2 - H 𝐰 vex , χ 3 , n + 1 , χ 4 ) ⁢ d n + 1 ⁢ 𝐰 , d n + 1 ⁢ 𝐰 〉

- 〈 ∂ θ ⁢ θ ⁡ ψ ^ n , ξ ^ 1 , n ⁢ d n + 1 ⁢ θ ^ , d n + 1 ⁢ θ ^ 〉 - 〈 ( H ^ 𝐰 cav , χ ^ 1 , n + 1 , χ ^ 2 - H ^ 𝐰 vex , χ ^ 3 , n + 1 , χ ^ 4 ) ⁢ d n + 1 ⁢ 𝐰 ^ , d n + 1 ⁢ 𝐰 ^ 〉

for

ξ 1 ∈ ( θ n , θ n + 1 ) , χ 1 , χ 3 ∈ ( ρ n , ρ n + 1 ) , χ 2 , χ 4 ∈ ( η n , η n + 1 ) ,

ξ ^ 1 ∈ ( θ ^ n , θ ^ n + 1 ) , χ ^ 1 ⁢ χ ^ 3 ∈ ( ρ ^ n , ρ ^ n + 1 ) , χ ^ 2 , χ ^ 4 ∈ ( η ^ n , η ^ n + 1 ) .

This can be further rearranged as follows:

2 ⁢ ( (i) + (ii) ) = 〈 ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , x ⁢ d n + 1 ⁢ ( θ - θ ^ ) , d n + 1 ⁢ ( θ - θ ^ ) 〉

- 〈 ( H 𝐰 vex , χ 3 , n + 1 , χ 4 - H 𝐰 cav , χ 1 , n + 1 , χ 2 ) ⁢ d n + 1 ⁢ ( 𝐰 - 𝐰 ^ ) , d n + 1 ⁢ ( 𝐰 - 𝐰 ^ ) 〉

- 〈 d n + 1 ⁢ θ ^ , ∂ θ ⁢ θ ⁡ ψ ^ n , ξ ^ 1 , n ⁢ d n + 1 ⁢ θ ^ + ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , n ⁢ d n + 1 ⁢ θ ^ - 2 ⁢ ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , n ⁢ d n + 1 ⁢ θ 〉

- 〈 d n + 1 ⁢ 𝐰 ^ , H ^ 𝐰 cav , n + 1 , χ ^ 2 ⁢ d n + 1 ⁢ 𝐰 ^ + H 𝐰 cav , χ 1 , n + 1 , χ 2 ⁢ d n + 1 ⁢ 𝐰 ^ - 2 ⁢ H 𝐰 cav , χ 1 , n + 1 , χ 2 ⁢ d n + 1 ⁢ 𝐰 〉

+ 〈 d n + 1 ⁢ 𝐰 ^ , H ^ 𝐰 vex , χ ^ 3 , n + 1 , χ ^ 4 ⁢ d n + 1 ⁢ 𝐰 ^ + H 𝐰 vex , χ 3 , n + 1 , χ 4 ⁢ d n + 1 ⁢ 𝐰 ^ - 2 ⁢ H 𝐰 vex , χ 3 , n + 1 , χ 4 ⁢ d n + 1 ⁢ 𝐰 〉 .

For the remaining terms we find by adding suitable zeros

(iii) + … + (xii) = 〈 ∂ θ ψ ( ρ n , θ n , η n | ρ ^ n , θ ^ n , η ^ n ) , d n + 1 θ ^ 〉 + 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ η ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n , d n + 1 η ^ 〉

+ 〈 ∂ θ ⁢ θ ⁡ ψ ^ n ⁢ ( θ n - θ ^ n ) + ∂ θ ⁢ ρ ⁡ ψ ^ n ⁢ ( ρ n - ρ ^ n ) + ∂ θ ⁢ η ⁡ ψ ^ ⁢ ( η n - η ^ n ) , d n + 1 ⁢ θ ^ 〉

+ 〈 ∂ ρ ⁢ θ ψ ^ vex n + 1 ( θ n + 1 - θ ^ n + 1 ) + ∂ ρ ⁢ θ ψ ^ cav n , n + 1 , n ( θ n + 1 - θ ^ n + 1 ) + ∂ ρ ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 )

+ ∂ ρ ⁢ ρ ψ ^ cav n , n + 1 , n ( ρ n - ρ ^ n ) + ∂ ρ ⁢ η ψ ^ vex n + 1 ( η n + 1 - η ^ n + 1 ) + ∂ ρ ⁢ η ψ ^ cav n , n + 1 , n ( η n - η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ η ⁢ θ ψ ^ n + 1 ( θ n + 1 - θ ^ n + 1 ) + ∂ η ⁢ θ ψ ^ n , n + 1 , n ( θ n + 1 - θ ^ n + 1 ) + ∂ η ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 )

+ ∂ η ⁢ ρ ψ ^ cav n , n + 1 , n ( ρ n - ρ ^ n ) + ∂ η ⁢ η ψ ^ vex n + 1 ( η n + 1 - η ^ n + 1 ) + ∂ η ⁢ η ψ ^ cav n , n + 1 , n ( η n - η ^ n ) , d n + 1 η ^ 〉

- 〈 ∂ θ ⁡ ψ ^ n + 1 - ∂ θ ⁡ ψ ^ n , θ n + 1 - θ ^ n + 1 〉

+ 〈 ∂ ρ ⁡ ψ ~ ⁢ ( ρ ^ , θ ^ n + 1 , η ^ ) , d n + 1 ⁢ ( ρ - ρ ^ ) 〉 + 〈 ∂ η ⁡ ψ ~ ⁢ ( ρ ^ , θ ^ n + 1 , η ^ ) , d n + 1 ⁢ ( η - η ^ ) 〉

- 〈 ∂ ρ ⁡ ψ ^ n + 1 , ρ n + 1 - ρ ^ n + 1 〉 + 〈 ∂ ρ ⁡ ψ ^ n , ρ n - ρ ^ n 〉 - 〈 ∂ η ⁡ ψ ^ n + 1 , η n + 1 - η ^ n + 1 〉 + 〈 ∂ η ⁡ ψ ^ n , η n - η ^ n 〉 .

The first three inner products are already part of the results. Hence, we concentrate on the rest. For the next step, we perform the following Taylor expansion and algebraic manipulation:

- 〈 ∂ θ ⁡ ψ ^ n + 1 - ∂ θ ⁡ ψ ^ n , θ n + 1 - θ ^ n + 1 〉 = - 〈 ∂ θ ⁢ θ ⁡ ψ ^ ω ^ 1 ⁢ d n + 1 ⁢ θ ^ + ∂ θ ⁢ ρ ⁡ ψ ^ ω ^ 1 ⁢ d n + 1 ⁢ ρ ^ + ∂ θ ⁢ η ⁡ ψ ^ ω ^ 1 ⁢ d n + 1 ⁢ η ^ , θ n + 1 - θ ^ n + 1 〉 , ω ^ 1 ∈ ( θ ^ n , θ ^ n + 1 ) ,

and

〈 ∂ ρ ⁡ ψ ~ ⁢ ( ρ ^ , θ ^ n + 1 , η ^ ) , d n + 1 ⁢ ( ρ - ρ ^ ) 〉 - 〈 ∂ ρ ⁡ ψ ^ n + 1 , ρ n + 1 - ρ ^ n + 1 〉 + 〈 ∂ ρ ⁡ ψ ^ n , ρ n - ρ ^ n 〉

= 〈 ∂ ρ ⁡ ψ ~ ⁢ ( ρ ^ , θ ^ n + 1 , η ^ ) - ∂ ρ ⁡ ψ ^ n + 1 , d n + 1 ⁢ ( ρ - ρ ^ ) 〉 + 〈 ∂ ρ ⁡ ψ ^ n - ∂ ρ ⁡ ψ ^ n + 1 , ρ n - ρ ^ n 〉 ,

〈 ∂ η ⁡ ψ ~ ⁢ ( ρ ^ , θ ^ n + 1 , η ^ ) , d n + 1 ⁢ ( η - η ^ ) 〉 - 〈 ∂ η ⁡ ψ ^ n + 1 , η n + 1 - η ^ n + 1 〉 + 〈 ∂ η ⁡ ψ ^ , ⁢ η n - η ^ n 〉

= 〈 ∂ η ⁡ ψ ~ ⁢ ( ρ ^ , θ n + 1 , η ^ ) - ∂ η ⁡ ψ ^ n + 1 , d n + 1 ⁢ ( η - η ^ ) 〉 + 〈 ∂ η ⁡ ψ n - ∂ η ⁡ ψ ^ n + 1 , η n - η ^ n 〉 .

With these insights, some rearrangement and expansion of the last two equalities we obtain

(iii) + … + (xii) = 〈 ∂ θ ψ ( ρ n , θ n , η n | ρ ^ n , θ ^ n , η ^ n ) , d n + 1 θ ^ 〉 + 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n , d n + 1 η ^ 〉

- 〈 ∂ θ ⁢ θ ⁡ ψ ^ n ⁢ d n + 1 ⁢ ( θ - θ ^ ) + ( ∂ θ ⁢ θ ⁡ ψ ^ ω ^ 1 - ∂ θ ⁢ θ ⁡ ψ ^ n ) ⁢ ( θ n + 1 - θ ^ n + 1 ) , d n + 1 ⁢ θ ^ 〉

+ 〈 ∂ θ ⁢ ρ ⁡ ψ ^ n ⁢ ( ρ n - ρ ^ n ) + ∂ θ ⁢ η ⁡ ψ ^ n ⁢ ( η n - η ^ n ) , d n + 1 ⁢ θ ^ 〉

- 〈 ∂ θ ⁢ ρ ⁡ ψ ^ ω ^ 1 ⁢ d n + 1 ⁢ ρ ^ n + 1 + ∂ θ ⁢ η ⁡ ψ ^ ω ^ 1 ⁢ d n + 1 ⁢ η ^ , θ n + 1 - θ ^ n + 1 〉

+ 〈 ∂ ρ ⁢ θ ψ ^ vex n + 1 ( θ n + 1 - θ ^ n + 1 ) + ∂ ρ ⁢ θ ψ ^ cav n , n + 1 , n ( θ n + 1 - θ ^ n + 1 ) + ∂ ρ ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 )

+ ∂ ρ ⁢ ρ ψ ^ cav n ( ρ n - ρ ^ n ) + ∂ ρ ⁢ η ψ ^ vex n + 1 ( η n + 1 - η ^ n + 1 ) + ∂ ρ ⁢ η ψ ^ cav n ( η n - η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ η ⁢ θ ψ ^ vex n + 1 ( θ n + 1 - θ ^ n + 1 ) + ∂ η ⁢ θ ψ ^ cav n , n + 1 , n ( θ n + 1 - θ ^ n + 1 ) + ∂ η ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 )

+ ∂ η ⁢ ρ ψ ^ cav n ( ρ n - ρ ^ n ) + ∂ η ⁢ η ψ ^ vex n + 1 ( η n + 1 - η ^ n + 1 ) + ∂ η ⁢ η ψ ^ cav n ( η n - η ^ n ) , d n + 1 η ^ 〉

- 〈 ∂ ρ ⁢ ρ ⁡ ψ ^ cav ω ^ 2 ⁢ d n + 1 ⁢ ρ ^ + ∂ ρ ⁢ η ⁡ ψ ^ cav ω ^ 2 ⁢ d n + 1 ⁢ η ^ , d n + 1 ⁢ ( ρ - ρ ^ ) 〉

- 〈 ∂ η ⁢ ρ ⁡ ψ ^ cav ω ^ 3 ⁢ d n + 1 ⁢ ρ ^ + ∂ η ⁢ η ⁡ ψ ^ cav ω ^ 3 ⁢ d n + 1 ⁢ η ^ , d n + 1 ⁢ ( η - η ^ ) 〉

- 〈 ∂ ρ ⁢ θ ⁡ ψ ^ ω ^ 4 ⁢ d n + 1 ⁢ θ ^ + ∂ ρ ⁢ ρ ⁡ ψ ^ ω ^ 4 ⁢ d n + 1 ⁢ ρ ^ + ∂ ρ ⁢ η ⁡ ψ ^ ω ^ 4 ⁢ d n + 1 ⁢ η ^ , ρ n - ρ ^ n 〉

- 〈 ∂ η ⁢ θ ⁡ ψ ω ^ 5 ⁢ d n + 1 ⁢ θ ^ + ∂ η ⁢ ρ ⁡ ψ ^ ω ^ 5 ⁢ d n + 1 ⁢ ρ ^ + ∂ η ⁢ η ⁡ ψ ^ ω ^ 5 ⁢ d n + 1 ⁢ η ^ , η n - η ^ n 〉

for

ω 2 , ω 3 ∈ [ ( ρ ^ n , η ^ n ) ⊤ , ( ρ ^ n + 1 , η ^ n + 1 ) ⊤ ]

and

ω 4 , ω 5 ∈ [ ( ρ ^ n , θ ^ n , η ^ n ) ⊤ , ( ρ ^ n + 1 , θ ^ n + 1 , η ^ n + 1 ) ⊤ ] .

We are now in the position to regroup the terms into the final form. Algebraic manipulation yields

(iii) - (xii) = 〈 ∂ θ ψ ( ρ n , θ n , η n | ρ ^ n , θ ^ n , η ^ n ) , d n + 1 θ ^ 〉 + 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n , d n + 1 η ^ 〉

- 〈 ∂ θ ⁢ θ ⁡ ψ ^ n ⁢ d n + 1 ⁢ ( θ - θ ^ ) + ( ∂ θ ⁢ θ ⁡ ψ ^ ω ^ 1 - ∂ θ ⁢ θ ⁡ ψ ^ n ) ⁢ ( θ n + 1 - θ ^ n + 1 ) , d n + 1 ⁢ θ ^ 〉

+ 〈 ( ∂ ρ ⁢ θ ⁡ ψ ^ n - ∂ ρ ⁢ θ ⁡ ψ ^ ω ^ 4 ) ⁢ ( ρ n - ρ ^ n ) + ( ∂ η ⁢ θ ⁡ ψ ^ n - ∂ η ⁢ θ ⁡ ψ ^ ω ^ 5 ) ⁢ ( η n - η ^ n ) , d n + 1 ⁢ θ ^ 〉

+ 〈 ( ∂ ρ ⁢ θ ⁡ ψ ^ vex n + 1 + ∂ ρ ⁢ θ ⁡ ψ ^ cav n , n + 1 , n - ∂ ρ ⁢ θ ⁡ ψ ^ ω ^ 1 ) ⁢ d n + 1 ⁢ ρ ^ , θ n + 1 - θ ^ n + 1 〉

+ 〈 ( ∂ η ⁢ θ ⁡ ψ ^ vex n + 1 + ∂ η ⁢ θ ⁡ ψ ^ cav n , n + 1 , n - ∂ η ⁢ θ ⁡ ψ ^ ω ^ 1 ) ⁢ d n + 1 ⁢ η ^ , θ n + 1 - θ ^ n + 1 〉

+ 〈 ∂ ρ ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 ) + ∂ ρ ⁢ ρ ψ ^ cav n , n + 1 , n ( ρ n - ρ ^ n ) + ∂ ρ ⁢ η ψ ^ vex n + 1 ( η n + 1 - η ^ n + 1 )

+ ∂ ρ ⁢ η ψ ^ cav n , n + 1 , n ( η n - η ^ n ) - ∂ ρ ⁢ ρ ψ ^ cav ω ^ 2 d n + 1 ( ρ - ρ ^ ) 〉 - ∂ ρ ⁢ η ψ ~ cav ω ^ 2 d n + 1 ( η - η ^ )

- ∂ ρ ⁢ ρ ψ ω ^ 4 ( ρ n - ρ ^ n ) - ∂ ρ ⁢ η ψ ω ^ 5 ( η n - η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ η ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 ) + ∂ η ⁢ ρ ψ ^ cav n , n + 1 , n ( ρ n - ρ ^ n ) + ∂ η ⁢ η ψ ^ vex n + 1 ( η n + 1 - η ^ n + 1 )

+ ∂ η ⁢ η ψ ^ cav n , n + 1 , n ( η n - η ^ n ) - ∂ ρ ⁢ η ψ ~ cav ω ^ 2 d n + 1 ( ρ - ρ ^ ) 〉 - ∂ η ⁢ η ψ ~ cav ω ^ 2 d n + 1 ( η - η ^ )

- ∂ ρ ⁢ η ψ ω ^ 4 ( ρ n - ρ ^ n ) - ∂ η ⁢ η ψ ω ^ 5 ( η n - η ^ n ) , d n + 1 η ^ 〉

= (a) + … + (f) .

We consider the last two inner products (e), (f) and by adding a suitable zero, we find

(e) + (f) = 〈 ∂ ρ ⁢ ρ ψ ^ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 ) + ∂ ρ ⁢ ρ ψ ^ cav n , n + 1 , n ( ρ n - ρ ^ n ) + ∂ ρ ⁢ η ψ vex n + 1 ( η n + 1 - η ^ n + 1 ) + ∂ ρ ⁢ η ψ ^ cav n , n + 1 , n ( η n - η ^ n )

- ∂ ρ ⁢ ρ ψ ^ cav ω ^ 2 d n + 1 ( ρ - ρ ^ ) 〉 - ∂ ρ ⁢ η ψ ^ cav ω ^ 2 d n + 1 ( η - η ^ ) - ∂ ρ ⁢ ρ ψ ^ ω ^ 4 ( ρ n - ρ ^ n )

- ∂ ρ ⁢ η ψ ^ ω ^ 5 ( η n - η ^ n ) , d n + 1 ρ ^ 〉

+ 〈 ∂ η ⁢ ρ ψ vex n + 1 ( ρ n + 1 - ρ ^ n + 1 ) + ∂ η ⁢ ρ ψ cav n , n + 1 , n ( ρ n - ρ ^ n ) + ∂ η ⁢ η ψ vex n + 1 ( η n + 1 - η ^ n + 1 )

+ ∂ η ⁢ η ψ cav n , n + 1 , n ( η n - η ^ n ) - ∂ ρ ⁢ η ψ cav ω ^ 2 d n + 1 ( ρ - ρ ^ ) 〉 - ∂ η ⁢ η ψ cav ω ^ 2 d n + 1 ( η - η ^ )

- ∂ ρ ⁢ η ψ ^ ω ^ 4 ( ρ n - ρ ^ n ) - ∂ η ⁢ η ψ ^ ω ^ 5 ( η n - η ^ n ) , d n + 1 η ^ 〉

= 〈 ∂ ρ ⁢ ρ ψ ^ vex n + 1 ( ρ - ρ ^ ) + ∂ ρ ⁢ η ψ ^ vex n + 1 d n + 1 ( η - η ^ ) + [ ∂ ρ ⁢ ρ ψ ^ vex n + 1 + ∂ ρ ⁢ ρ ψ ^ cav n , n + 1 , n - ∂ ρ ⁢ ρ ψ ^ ω ^ 4 ] ( ρ n - ρ ^ n )

+ [ ∂ ρ ⁢ η ⁡ ψ ^ vex n + 1 + ∂ ρ ⁢ η ⁡ ψ ^ cav n , n + 1 , n - ∂ ρ ⁢ η ⁡ ψ ^ ω ^ 5 ] ⁢ ( η n - η ^ n ) - ∂ ρ ⁢ ρ ⁡ ψ ^ cav ω ^ 2 ⁢ d n + 1 ⁢ ( ρ - ρ ^ )

- ∂ ρ ⁢ η ψ ^ cav ω ^ 2 d n + 1 ( η - η ^ ) , d n + 1 ρ ^ 〉

+ 〈 ∂ ρ ⁢ η ψ ^ vex n + 1 d n + 1 ( ρ - ρ ^ ) + ∂ η ⁢ η ψ ^ vex n + 1 d n + 1 ( η - η ^ ) + [ ∂ ρ ⁢ η ψ ^ vex n + 1 + ∂ ρ ⁢ η ψ ^ cav n , n + 1 , n - ∂ ρ ⁢ η ψ ^ ω ^ 4 ] ( ρ n - ρ ^ n )

+ [ ∂ η ⁢ η ⁡ ψ ^ vex n + 1 + ∂ η ⁢ η ⁡ ψ ^ cav n , n + 1 , n - ∂ η ⁢ η ⁡ ψ ^ ω ^ 5 ] ⁢ ( η n - η ^ n ) - ∂ η ⁢ ρ ⁡ ψ ^ cav ω ^ 2 ⁢ d n + 1 ⁢ ( ρ - ρ ^ )

- ∂ η ⁢ η ψ ^ cav ω ^ 2 d n + 1 ( η - η ^ ) , d n + 1 η ^ 〉

≤ 〈 ( H 𝐰 vex , n + 1 - H 𝐰 cav , ω ^ 2 ) ⁢ d n + 1 ⁢ ( 𝐰 - 𝐰 ^ ) , d n + 1 ⁢ 𝐰 ^ 〉

+ τ ⁢ ( ∥ ρ n - ρ ^ n ∥ 0 + ∥ η n - η ^ n ∥ 0 ) ⁢ ∥ d τ n + 1 ⁢ ρ ^ ∥ 0 , ∞ ⁢ ( ∥ 𝐮 ^ n + 1 - ω 4 ∥ 0 , ∞ + ∥ 𝐮 ^ n - ω 4 ∥ 0 , ∞ )

+ τ ⁢ ( ∥ ρ n - ρ ^ n ∥ 0 + ∥ η n - η ^ n ∥ 0 ) ⁢ ∥ d τ n + 1 ⁢ η ^ ∥ 0 , ∞ ⁢ ( ∥ 𝐮 ^ n + 1 - ω 5 ∥ 0 , ∞ + ∥ 𝐮 ^ n - ω 5 ∥ 0 , ∞ ) .

Here we used the notation 𝐮 = ( ρ , θ , η ) ⊤ . Similarly, we estimate

(a) + … + (d) = - 〈 ∂ θ ⁢ θ ⁡ ψ ^ n ⁢ d n + 1 ⁢ ( θ - θ ^ ) + ( ∂ θ ⁢ θ ⁡ ψ ^ ω ^ 1 - ∂ θ ⁢ θ ⁡ ψ ^ n ) ⁢ ( θ n + 1 - θ ^ n + 1 ) , d n + 1 ⁢ θ ^ 〉

+ 〈 ( ∂ ρ ⁢ θ ⁡ ψ ^ n - ∂ ρ ⁢ θ ⁡ ψ ^ ω ^ 4 ) ⁢ ( ρ n - ρ ^ n ) + ( ∂ η ⁢ θ ⁡ ψ ^ n - ∂ η ⁢ θ ⁡ ψ ^ ω ^ 5 ) ⁢ ( η n - η ^ n ) , d n + 1 ⁢ θ ^ 〉

≤ - 〈 ∂ θ ⁢ θ ψ ^ n d n + 1 ( θ - θ ^ ) , d n + 1 θ ^ 〉 + τ ∥ d τ n + 1 θ ^ ∥ 0 , ∞ ( ∥ θ n + 1 - θ ^ n + 1 ∥ 0 2 ∥ 𝐮 ^ n - ω 1 ∥ 0 , ∞

+ ∥ ρ n - ρ ^ n ∥ 0 2 ∥ 𝐮 ^ n - ω 4 ∥ 0 , ∞ + ∥ η n - η ^ n ∥ 0 2 ∥ 𝐮 ^ n - ω 5 ∥ 0 , ∞ ) .

In the following we will combine this with (i), (ii) and after rearrangement we find

ℛ ≤ - 𝒟 num n + 1 ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) + 〈 ∂ θ ψ ( ρ n , θ n , η | ρ ^ n , θ ^ n , η ^ n ) , d n + 1 θ ^ 〉

+ 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n ) , d n + 1 ρ ^ 〉 + 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n , d n + 1 η ^ 〉

+ 1 2 ⁢ 〈 d n + 1 ⁢ θ ^ , 2 ⁢ ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , n ⁢ d n + 1 ⁢ θ - 2 ⁢ ∂ θ ⁢ θ ⁡ ψ ^ n ⁢ d n + 1 ⁢ ( θ - θ ^ ) - ∂ θ ⁢ θ ⁡ ψ ^ n , ξ ^ 1 , n ⁢ d n + 1 ⁢ θ ^ - ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , n ⁢ d n + 1 ⁢ θ ^ 〉

+ 1 2 〈 d n + 1 𝐰 ^ , 2 H 𝐰 cav , χ 1 , n + 1 , χ 2 d n + 1 𝐰 - 2 H 𝐰 cav , ω ^ 2 d n + 1 ( 𝐰 - 𝐰 ^ ) - H ^ 𝐰 cav , χ ^ 1 , n + 1 , χ ^ 2 d n + 1 𝐰 ^ - H 𝐰 cav , χ 1 , n + 1 , χ 2 ) d n + 1 𝐰 ^ 〉

- 1 2 ⁢ 〈 d n + 1 ⁢ 𝐰 ^ , 2 ⁢ H 𝐰 vex , χ 3 , n + 1 , χ 4 ⁢ d n + 1 ⁢ 𝐰 - 2 ⁢ H ^ 𝐰 vex , n + 1 ⁢ d n + 1 ⁢ ( 𝐰 - 𝐰 ^ ) - H ^ 𝐰 vex ( χ ^ 3 , n + 1 , χ ^ 4 ⁢ d n + 1 ⁢ 𝐰 ^ - H 𝐰 vex , χ 3 , n + 1 , χ 4 ⁢ d n + 1 ⁢ 𝐰 ^ 〉

+ τ ⁢ ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ⁢ ( ∥ θ n + 1 - θ ^ n + 1 ∥ 0 2 ⁢ ∥ 𝐮 ^ n - ω 1 ∥ 0 , ∞ + ∥ ρ n - ρ ^ n ∥ 0 2 ⁢ ∥ 𝐮 ^ n - ω 4 ∥ 0 , ∞ + ∥ η n - η ^ n ∥ 0 2 ⁢ ∥ 𝐮 ^ n - ω 5 ∥ 0 , ∞ )

+ τ ⁢ ( ∥ ρ n - ρ ^ n ∥ 0 + ∥ η n - η ^ n ∥ 0 ) ⁢ ∥ d τ n + 1 ⁢ ρ ^ ∥ 0 , ∞ ⁢ ( ∥ 𝐮 ^ n + 1 - ω 4 ∥ 0 , ∞ + ∥ 𝐮 ^ n - ω 4 ∥ 0 , ∞ )

+ τ ⁢ ( ∥ ρ n - ρ ^ n ∥ 0 + ∥ η n - η ^ n ∥ 0 ) ⁢ ∥ d τ n + 1 ⁢ η ^ ∥ 0 , ∞ ⁢ ( ∥ 𝐮 ^ n + 1 - ω 5 ∥ 0 , ∞ + ∥ 𝐮 ^ n - ω 5 ∥ 0 , ∞ )

≤ 𝒟 num n + 1 ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) + 〈 ∂ θ ψ ( ρ n , θ n , η | ρ ^ n , θ ^ n , η ^ n ) , d n + 1 θ ^ 〉

+ 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n ) , d n + 1 ρ ^ 〉 + 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n , d n + 1 η ^ 〉

+ τ ⁢ ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ⁢ ∥ d n + 1 ⁢ θ ∥ 0 ⁢ ∥ ( ρ n , ξ 1 , η n ) ⊤ - 𝐮 ^ n ∥ 0

+ τ 2 ⁢ ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ 2 ⁢ ( ∥ ( ρ ^ n , ξ ^ 1 , η ^ n ) ⊤ - 𝐮 ^ n ∥ 0 + ∥ ( ρ n , ξ 1 , η n ) ⊤ - 𝐮 ^ n ∥ 0 )

+ τ ∥ d τ n + 1 𝐰 ^ ∥ 0 , ∞ ∥ d n + 1 𝐰 ∥ 0 ∥ ( χ 1 , χ 2 ) ) ⊤ - ω ^ 2 ∥ 0 + τ 2 ∥ d τ n + 1 𝐰 ^ ∥ 0 , ∞ 2 ( ∥ ( χ 1 , χ 2 ) ⊤ - ω ^ 2 ∥ 0 + ∥ ( χ ^ 1 , χ ^ 2 ) ⊤ - ω ^ 2 ∥ 0 )

+ τ ⁢ ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 , ∞ 2 ⁢ ∥ ( χ 3 , θ n + 1 , χ 4 ) ⊤ - 𝐮 ^ n + 1 ∥ 0

+ τ 2 ⁢ ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 , ∞ 2 ⁢ ( ∥ ( χ 3 , θ n + 1 , χ 4 ) ⊤ - 𝐮 ^ n + 1 ∥ 0 + ∥ ( χ ^ 3 , θ ^ n + 1 , χ ^ 4 ) ⊤ - 𝐮 ^ n + 1 ∥ 0 )

+ τ ⁢ ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ⁢ ( ∥ θ n + 1 - θ ^ n + 1 ∥ 0 2 ⁢ ∥ θ ^ n - ω ^ 1 ∥ 0 , ∞ + ∥ ρ n - ρ ^ n ∥ 0 2 ⁢ ∥ 𝐮 ^ n - ω ^ 4 ∥ 0 , ∞ + ∥ η n - η ^ n ∥ 0 2 ⁢ ∥ 𝐮 ^ n - ω ^ 5 ∥ 0 , ∞ )

+ τ ⁢ ( ∥ ρ n - ρ ^ n ∥ 0 + ∥ η n - η ^ n ∥ 0 ) ⁢ ∥ d τ n + 1 ⁢ ρ ^ ∥ 0 , ∞ ⁢ ( ∥ 𝐮 ^ n + 1 - ω ^ 4 ∥ 0 , ∞ + ∥ 𝐮 ^ n - ω ^ 4 ∥ 0 , ∞ )

+ τ ⁢ ( ∥ ρ n - ρ ^ n ∥ 0 + ∥ η n - η ^ n ∥ 0 ) ⁢ ∥ d τ n + 1 ⁢ η ^ ∥ 0 , ∞ ⁢ ( ∥ 𝐮 ^ n + 1 - ω ^ 5 ∥ 0 , ∞ + ∥ 𝐮 ^ n - ω ^ 5 ∥ 0 , ∞ )

with the numerical dissipation

𝒟 num n + 1 ⁢ ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) = γ ρ 2 ⁢ ∥ ∇ ⁡ d n + 1 ⁢ ( ρ - ρ ^ ) ∥ 0 2 + γ η 2 ⁢ ∥ ∇ ⁡ d n + 1 ⁢ ( η - η ^ ) ∥ 0 2 + 〈 ∂ θ ⁢ θ ⁡ ψ n , ξ 1 , x ⁢ d n + 1 ⁢ ( θ - θ ^ ) , d n + 1 ⁢ ( θ - θ ^ ) 〉

- 〈 ( H 𝐰 vex , χ 3 , n + 1 , χ 4 - H 𝐰 cav , χ 1 , n + 1 , χ 2 ) ⁢ d n + 1 ⁢ ( 𝐰 - 𝐰 ^ ) , d n + 1 ⁢ ( 𝐰 - 𝐰 ^ ) 〉 ≤ 0 .

We estimate all remaining norms, which by construction yields

∥ d n + 1 ⁢ θ ∥ 0 ≤ ∥ θ n + 1 - θ ^ n + 1 ∥ 0 + ∥ θ n - θ ^ n ∥ 0 + τ ⁢ ∥ d τ n + 1 ⁢ θ ^ ∥ 0 ,

∥ ( ρ n , ξ 1 , η n ) ⊤ - 𝐮 ^ n ∥ 0 ≤ ∥ 𝐰 n - 𝐰 ^ n ∥ 0 + C ⁢ ∥ θ n - θ ^ n ∥ 0 + C ⁢ ∥ θ n + 1 - θ ^ n + 1 ∥ 0 + τ ⁢ ∥ d τ n + 1 ⁢ θ ^ n + 1 ∥ 0 ,

∥ ( ρ ^ n , ξ ^ 1 , η ^ n ) ⊤ - 𝐮 ^ n ∥ 0 ≤ C ⁢ τ ⁢ ∥ d τ n + 1 ⁢ θ ^ ∥ 0 ,

∥ d n + 1 ⁢ 𝐰 ∥ 0 ≤ ∥ 𝐰 n + 1 - 𝐰 ^ n + 1 ∥ 0 + ∥ 𝐰 n - 𝐰 ^ n ∥ 0 + τ ⁢ ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 ,

∥ ( χ 1 , χ 2 ) ) ⊤ - ω ^ 2 ∥ 0 ≤ C ( ∥ 𝐰 n - 𝐰 ^ n ∥ 0 + ∥ 𝐰 n + 1 - 𝐰 ^ n + 1 ∥ 0 + τ ∥ d τ n + 1 𝐰 ^ ∥ 0 ) ,

∥ ( χ ^ 1 , χ ^ 2 ) ⊤ - ω ^ 2 ∥ 0 ≤ C ⁢ τ ⁢ ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 2 ,

∥ ( χ 3 , θ n + 1 , χ 4 ) ⊤ - 𝐮 ^ n + 1 ∥ 0 ≤ C ⁢ ( ∥ 𝐰 n - 𝐰 ^ n ∥ 0 + ∥ 𝐰 n + 1 - 𝐰 ^ n + 1 ∥ 0 + τ ⁢ ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 ) ,

∥ ( χ ^ 3 , θ ^ n + 1 , χ ^ 4 ) ⊤ - 𝐮 ^ n + 1 ∥ 0 ≤ C ⁢ τ ⁢ ( ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 + ∥ d τ n + 1 ⁢ θ ^ ∥ 0 ) ,

∥ θ ^ n - ω ^ 1 ∥ 0 , ∞ ≤ C ⁢ τ ⁢ ∥ d τ n + 1 ⁢ θ ^ n + 1 ∥ 0 , ∞ ,

∥ 𝐮 ^ n - ω ^ 4 ∥ 0 , ∞ ≤ C ⁢ τ ⁢ ( ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 , ∞ + ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ) ,

∥ 𝐮 ^ n - ω ^ 5 ∥ 0 , ∞ ≤ C ⁢ τ ⁢ ( ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 , ∞ + ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ) ,

∥ 𝐮 ^ n + 1 - ω ^ 4 ∥ 0 , ∞ ≤ C ⁢ τ ⁢ ( ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 , ∞ + ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ) ,

∥ 𝐮 ^ n + 1 - ω ^ 5 ∥ 0 , ∞ ≤ C ⁢ τ ⁢ ( ∥ d τ n + 1 ⁢ 𝐰 ^ ∥ 0 , ∞ + ∥ d τ n + 1 ⁢ θ ^ ∥ 0 , ∞ ) .

Using the above estimates yields

ℛ ≤ - 𝒟 num n + 1 ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) + 〈 ∂ θ ψ ( ρ n , θ n , η | ρ ^ n , θ ^ n , η ^ n ) , d n + 1 θ ^ 〉

+ 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n ) , d n + 1 ρ ^ 〉 + 〈 ∂ θ ψ ~ ( ρ , θ n + 1 , η | ρ ^ , θ ^ n + 1 , η ^ n , d n + 1 η ^ 〉

+ C τ ( 𝒲 ( ρ , θ , η | ρ ^ , θ ^ , η ^ ) n + 𝒲 ( ρ , θ , η | ρ ^ , θ ^ , η ^ ) n + 1 ) + C τ 3 .

The remaining terms are discrete versions of the remainder in the relative entropy ansatz. However, they can be estimated in a straightforward manner such that

ℛ ≤ - 𝒟 num n + 1 ⁢ ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) + C ⁢ τ ⁢ ( 𝒲 n + 𝒲 n + 1 ) + C ⁢ τ 3 .

A.3 Dissipative Contribution

Let us consider now the dissipative contributions, i.e.,

𝒟 = - 〈 d n + 1 ⁢ ( e - e ^ ) , θ n + 1 - θ ^ n + 1 〉 + 〈 d n + 1 ⁢ ( ρ - ρ ^ ) , μ ρ n + 1 - μ ^ ρ n + 1 + r 2 n + 1 〉

+ 〈 d n + 1 ⁢ ( η - η ^ ) , μ η n + 1 - μ ^ η n + 1 + r 5 n + 1 〉

Inserting into the v 1 = μ ρ n + 1 - μ ^ ρ n + 1 + r 2 n + 1 , ξ = θ n + 1 - θ ^ n + 1 , w 1 = μ η n + 1 - μ ^ η n + 1 + r 5 n + 1 into (8.1)–(8.5) and (8.6)–(8.10) yields

𝒟 = - 〈 𝐋 11 ⁢ ∇ ⁡ ( μ ρ n + 1 - μ ^ ρ n + 1 ) - 𝐋 12 ⁢ ∇ ⁡ ( θ n + 1 - θ ^ n + 1 ) + ( μ η n + 1 - μ ^ η n + 1 ) ⁢ 𝐋 13 , ∇ ⁡ ( μ ρ n + 1 - μ ^ ρ n + 1 + r 2 n + 1 ) 〉

+ 〈 𝐋 12 ∇ ( μ ρ n + 1 - μ ^ ρ n + 1 ) - 𝐋 22 ∇ ( θ n + 1 - θ ^ n + 1 ) + ( μ η n + 1 - μ ^ η n + 1 ) 𝐋 23 , ∇ ( θ n + 1 - θ ^ n + 1 〉

- 〈 𝐋 13 ⋅ ∇ ⁡ ( μ ρ n + 1 - μ ^ ρ n + 1 ) - 𝐋 23 ⋅ ∇ ⁡ ( θ n + 1 - θ ^ n + 1 ) + 𝐋 33 ⁢ ( μ η n + 1 - μ ^ η n + 1 ) , μ η n + 1 - μ ^ η n + 1 + r 5 n + 1 〉

+ 〈 r 1 n + 1 , μ ρ n + 1 - μ ^ ρ n + 1 + r 2 n + 1 〉 - 〈 r 3 n + 1 , θ h n + 1 - θ ^ h n + 1 〉 + 〈 r 4 n + 1 , μ η n + 1 - μ ^ η n + 1 + r 5 n + 1 〉

≤ - ( 1 - 2 ⁢ δ ) ⁢ τ ⁢ 𝒟 𝐋 ⁢ ( μ ρ n + 1 - μ ^ ρ n + 1 , θ n + 1 - θ ^ n + 1 , μ η n + 1 - μ ^ η n + 1 ) + C ⁢ τ ⁢ 〈 ψ ~ ρ - ψ ^ ~ ρ , 1 〉 2 + C ⁢ τ ⁢ ∥ θ n + 1 - θ ^ n + 1 ∥ 0 2

+ C ( 𝐋 ) τ ( ∥ r 1 n + 1 ∥ - 1 2 + ∥ r 2 n + 1 ∥ 1 2 + ∥ r 3 n + 1 ∥ - 1 2 + ∥ r 4 n + 1 ∥ 0 2 + ∥ r 5 n + 1 ∥ 0 2 .

A.4 Quadratic Part

Due to the non-convex nature, the relative entropy is stabilised by an L 2 term for ρ - ρ ^ , η - η ^ . The temporal change in time is easily computed as

λ 2 ⁢ ( ∥ ρ n + 1 - ρ ^ n + 1 ∥ 0 2 + ∥ η n + 1 - η ^ n + 1 ∥ 0 2 - ∥ ρ n - ρ ^ n ∥ 0 2 - ∥ η n - η ^ n ∥ 0 2 )

= λ ⁢ 〈 d n + 1 ⁢ ( ρ - ρ ^ ) , ρ n + 1 - ρ ^ n + 1 〉 + λ ⁢ 〈 d n + 1 ⁢ ( η - η ^ ) , η n + 1 - η ^ n + 1 〉 - λ 2 ⁢ ∥ d n + 1 ⁢ ( ρ - ρ ^ ) ∥ 0 2 - λ 2 ⁢ ∥ d n + 1 ⁢ ( η - η ^ ) ∥ 0 2

= (i) + (ii) + (iii) + (iv) .

For the first and second term we insert v 1 = ρ n + 1 - ρ ^ n + 1 into (8.1), (8.6) and w 1 = η n + 1 - η ^ n + 1 into (8.4), (8.9) which yields

(i) + (ii) = - λ ⁢ τ ⁢ 〈 𝐋 11 ⁢ ∇ ⁡ ( μ ρ n + 1 - μ ^ ρ n + 1 ) - 𝐋 12 ⁢ ∇ ⁡ ( θ n + 1 - θ ^ n + 1 ) + ( μ η n + 1 - μ ^ η n + 1 ) ⁢ 𝐋 13 , ∇ ⁡ ( ρ n + 1 - ρ ^ n + 1 ) 〉

- λ ⁢ τ ⁢ 〈 𝐋 13 ⋅ ∇ ⁡ ( μ ρ n + 1 - μ ^ ρ n + 1 ) - 𝐋 23 ⋅ ∇ ⁡ ( θ n + 1 - θ ^ n + 1 ) + 𝐋 33 ⁢ ( μ η n + 1 - μ ^ η n + 1 ) , η n + 1 - η ^ n + 1 〉

+ λ ⁢ τ ⁢ 〈 r 1 n + 1 , ρ n + 1 - ρ ^ n + 1 〉 + λ ⁢ τ ⁢ 〈 r 4 n + 1 , η n + 1 - η ^ n + 1 〉

≤ δ ⁢ τ ⁢ 𝒟 𝐋 ⁢ ( μ ρ n + 1 - μ ^ ρ n + 1 , θ n + 1 - θ ^ n + 1 , μ η n + 1 - μ ^ η n + 1 )

+ C ⁢ ( 𝐋 , λ ) ⁢ τ ⁢ ( ∥ ρ n + 1 - ρ ^ n + 1 ∥ 1 2 + ∥ η n + 1 - η ^ n + 1 ∥ 0 2 ) + C ⁢ τ ⁢ ( ∥ r 1 n + 1 ∥ - 1 2 + ∥ r 4 n + 1 ∥ 0 2 ) .

A.5 Summation of All Estimates

Here we will collect all estimates from above and recall Lemma 7, i.e., that the H 1 - norm of ρ n + 1 - ρ ^ n + 1 and η n + 1 - η ^ n + 1 can be bounded by the relative entropy. Summing the above results together and setting δ = 1 6 , we find

(A.1)

𝒲 λ ( ρ , θ , η | ρ ^ , θ ^ , η ) | t n t n + 1 + τ 2 𝒟 𝐋 ( μ ρ n + 1 - μ ^ ρ n + 1 , θ n + 1 - θ ^ n + 1 , μ η n + 1 - μ ^ η n + 1 ) + 𝒟 ~ num n + 1 ( ρ - ρ ^ , θ - θ ^ , η - η ^ )

≤ C τ 𝒲 λ ( ρ , θ , η | ρ ^ , θ ^ , η ) n + 1 + C τ 𝒲 λ ( ρ , θ , η | ρ ^ , θ ^ , η ) n + C τ 3

+ τ C ( 𝐋 ) ( ∥ r 1 n + 1 ∥ - 1 2 + ∥ r 2 n + 1 ∥ 1 2 + ∥ r 3 n + 1 ∥ - 1 2 + ∥ r 4 n + 1 ∥ 0 2 + ∥ r 5 n + 1 ∥ 0 2

with relative numerical dissipation

𝒟 ~ num n + 1 ⁢ ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) := 𝒟 num n + 1 ⁢ ( ρ - ρ ^ , θ - θ ^ , η - η ^ ) + λ 2 ⁢ ∥ d n + 1 ⁢ ( ρ - ρ ^ ) ∥ 0 2 + λ 2 ⁢ ∥ d n + 1 ⁢ ( η - η ^ ) ∥ 0 2 .

The final results follow from the discrete Gronwall lemma setting with τ sufficiently small.

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Received: 2023-12-22

Revised: 2024-04-15

Accepted: 2024-07-29

Published Online: 2024-09-03

Published in Print: 2025-04-01

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Articles in the same Issue

https://doi.org/10.1515/cmam-2023-0274

Keywords for this article

Non-Isothermal Phase-Field; Cross-Diffusion; Finite Elements; Entropy Stable