发表机构
Virginia Commonwealth University; The University of Tennessee, Knoxville; The University of Utah（弗吉尼亚联邦大学; 田纳西大学诺克斯维尔分校; 犹他大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对双参数扩散域方法（DDM2p）相关的线性退化界面传输问题，证明其正则化能量Γ-收敛到极限能量泛函，得到O(α)收敛速率，且数值实验验证了该一阶速率的尖锐性。
AI 中文摘要
我们研究了一类线性退化界面传输问题的奇异极限，这类问题源自新提出的双参数扩散域方法（DDM2p）中的正则化程序。对于α>0，正则化问题在H¹(Ω)上具有严格凸的变分形式。在α→0的极限下，该问题退化为具有非标准能量结构的弱耦合界面系统。为刻画该极限，我们引入了H¹(Ω)的闭希尔伯特子空间𝒽，其通过在Ω的环形子域Ω₂上的辅助亥姆霍兹问题定义，并确定了𝒽上的极限能量泛函ℰ₀。我们利用标准框架证明，正则化能量ℰ_α在强L²(Ω)拓扑下Γ-收敛到ℰ₀。因此，ℰ_α的极小值点收敛到ℰ₀的唯一极小值点，该极小值点被证明等价于极限界面问题的解。我们进一步证明u_α在H¹(Ω)中强收敛到u₀，并建立了O(α)的收敛速率。一维空间中的数值实验证实了所预测的一阶收敛速率，并表明该速率是尖锐的。
英文摘要
We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For $α>0$, the regularized problem admits a strictly convex variational formulation on $H^{1}(Ω)$. In the limit $α\to0$, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace $\mathcal{H}\subset H^{1}(Ω)$, defined through an auxiliary Helmholtz problem on an annular subdomain $Ω_2\subset Ω$, and identify the limiting energy functional $\mathcal{E}_{0}$ on $\mathcal{H}$. We prove that the regularized energies $\mathcal{E}_α$ $Γ$-converge to $\mathcal{E}_{0}$ in the strong $L^{2}(Ω)$ topology, using the standard framework. Consequently, minimizers of $\mathcal{E}_α$ converge to the unique minimizer of $\mathcal{E}_{0}$, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence $u_α\to u_{0}$ in $H^{1}(Ω)$ and establish an $O(α)$ convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.
