Two-parameter convergence of diffuse domain solutions

Determine whether the solutions \(u_{\varepsilon,\alpha}\) of the two-parameter diffuse domain problem converge to the limiting solution \(u_{0,0}\) as \((\varepsilon,\alpha)\to(0,0)\), specify the mode of convergence, and determine the convergence orders with respect to \(\alpha\) and \(\varepsilon\).

Background

The paper proves convergence in H1(Ω)H^1(\Omega) for the epsilon limit with fixed positive α\alpha, and proves an O(α)O(\alpha) rate for the subsequent alpha limit. It does not establish convergence when both parameters tend to zero simultaneously.

The unresolved issue includes both the correct topology or norm and quantitative error estimates in each parameter, which are needed to assess simultaneous refinement of the DDM2p approximation.

References

Is the following two-parameter convergence valid: ${u_{\varepsilon,\alpha}\longrightarrow u_{0,0}$, as $(\varepsilon,\alpha)\to (0,0)$? In what sense? At what orders with respect to $\alpha$ and $\varepsilon$?

— The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem  (2609.01237 - Luong et al., 1 Sep 2026) in Section 2, subsection “Convergence Questions,” Q6

Is the following convergence valid: ${u_{\varepsilon,\varepsilonp} \longrightarrow u_{0,0}$, as $\varepsilon\searrow 0$? In what sense? At what order with respect to $\varepsilon$?

— The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem  (2609.01237 - Luong et al., 1 Sep 2026) in Section 2, subsection “Convergence Questions,” Q7

Is the following formal convergence valid: ${u_{\varepsilon,\varepsilonp} \longrightarrow u_{0,0}$, asymptotically as $\varepsilon\searrow 0$? If so, at what order with respect to $\varepsilon$? This is a classic {\it corner layer} problem within the field of matched asymptotic analysis.

— The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem  (2609.01237 - Luong et al., 1 Sep 2026) in Section 2, subsection “Convergence Questions,” Q8