Dynamic flux surrogate-based partitioned methods for interface problems

The paper proves stability of the DMD-FS scheme (Theorem 2) by writing the one-step energy identity (their eq. (34)) and then expressing the interface energy-imbalance term Ek as a telescoping history involving S(λS−λ) (their eq. (41)), bounding the two resulting sums via lifting/trace/inverse estimates (Lemmas 2–4) and the DMD accuracy condition (39), to obtain the final bound with the same CFL scaling and a +CF T contribution (Theorem 2) . The candidate solution proves the same inequality by a different route: it keeps the interface term in the one-step energy identity and directly splits λk = λS k − ek, absorbing the jump term by trace/coercivity and bounding λS k via a Schur complement coercivity/spectral-equivalence and discrete liftings; the DMD-accuracy term is then summed over time to produce the same +CF T scaling. This approach is not the one used in the paper, and it invokes a standard Schur–complement spectral equivalence not explicitly stated in the manuscript, but under customary FE assumptions it yields the stated stability estimate as well. Hence, both are correct, with different proofs.