Q2. What is the key ingredient to obtain such a uniform stability?
The key ingredient, to obtain such an added-mass uniform stability, is the time penalty term on the fluid pressure fluctuations the authors propose to add on the interface.
Q3. What are the key ingredients for the stability of the method?
The key ingredients for the stability of the method are:• the Nitsche treatment of the interface coupling conditions,• the addition of a weakly consistent penalization of the (time) fluid pressure fluctuations at the interface.
Q4. What is the proof of Lemma 5.2?
The proof of Theorem 5.2 is based, exclusively, on the dissipation due to the Nitsche coupling and the time pressure penalization term.
Q5. Why is the order of the scheme expected to be O(h 1 2 t 1?
As regards accuracy, the order of the scheme is expected to be O(h 1 2 δt 1 2 ), due to the weak consistency of the stabilization term.
Q6. What is the simplest method of estimating the non-linear coupled problem?
The stabilized explicit scheme, with K ≥ 0 defect-correction iterations, applied to the non-linear coupled problem (49)-(51), is then given by the following iterative procedure.
Q7. What is the disadvantage of implicit coupling?
Despite the outstanding stability properties provided by the previous Lemma, implicit coupling has the major disadvantage of being too CPU-time consuming.
Q8. What is the corresponding error for the stabilized explicit coupling scheme?
for the stabilized explicit coupling scheme the convergence rate (in time) is expected to be O(δt 1 2 ), whereas for the implicit scheme an optimal O(δt) is assumed.
Q9. What is the error bound for the stabilization term?
Since the authors expect the error in time to be dominated by the contribution from the stabilization term, the error bound (41) should then take the formEn ≤ C [ E0 + ( γ0Tγµ) 1 2h 1 2 δt 1 2 ] . (44)In particular, since δt = O(h), the authors should haveEn ≤ C [ E0 + (CΣγ0T ) 1 2µ 1 2 γh ] . (45)RR n° 644522 E. Burman & M.A. Fernández•
Q10. How many times faster is the implicit coupling?
The authors notice that the explicit coupling is 8 times faster than the implicit coupling (involving an average of two Newton iterations per time step).
Q11. How can the implicit coupling scheme be stabilized?
In Section §5, the authors will show that the explicit coupling scheme can be stabilized by adding, to the fluid sub-problem, a suitable interface time-penalization term acting on the pressure.
Q12. What is the kinematic condition of the stabilized explicit coupling scheme?
−Qn+1IMPLICIT∣∣ max0≤n≤N−1 ∣∣Qn+1IMPLICIT∣∣ , of the stabilized explicit coupling scheme (K = 0) with respect to the implicit coupling scheme (strongly enforced kinematic condition).
Q13. What is the effect of the weak consistency of the stabilization term?
This illustrates the impact of the optimality loss introduced by weak consistency of the stabilization term, present in the error estimate (44) with a loss of half-a-power in δt.
Q14. What is the inverse of the Nitsche penalty coupling term?
By testing (21) with(vh, qh,wh, ẇh) = (un+1h , p n+1 h , ∂δtη n+1 h , ∂δtη̇ n+1 h ),using (22), multiplying by δt, replacing index n by m and summing over 0 ≤ m ≤ n − 1 and using the stability analysis of the implicit scheme (note that condition (23) implies (6)), the authors haveEn ≤E0 − γµ h δt n−1∑ m=0 ∫ Σ ( um+1h − umh ) · ∂δtηm+1h︸ ︷︷ ︸ T1− δt n−1∑ m=0 ∫ Σ ( σ(um+1h , p m+1 h )n− σ(umh , pmh )n ) · (um+1h − ∂δtηm+1h ) =E0 − T1 − δtn−1∑ m=0 ∫ Σ 2µ ( (um+1h )n− (umh )n ) · (um+1h − ∂δtηm+1h )︸ ︷︷ ︸ T2+ δt n−1∑ m=0 ∫ Σ(pm+1h − pmh )(um+1h − ∂δtηm+1h ) · n︸ ︷︷ ︸ T3 .(26)RR n° 644514 E. Burman & M.A. FernándezAs mentioned above, the term T1 involving the fluid velocity fluctuations at the interface can be handled using the Nitsche’s penalty coupling term.