arXiv subjects

Ming Hei Wong

Publications and source records attributed to Ming Hei Wong.

4 recordsLinked to original sources

arXiv2026

The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem

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.

math.AP↗

arXiv2026

The symmetric V-cycle can diverge under the multigrid axioms for cell-centred discretisations

The axiomatic convergence theory for multigrid methods applied to cell-centred finite-difference and finite-volume discretisations rests on two hypotheses: an imbalanced Galerkin condition (G3), which states that $R_{\ell-1}A_\ell P_{\ell-1}=2A_{\ell-1}$ with $R_{\ell-1}=\frac{1}{2}P_{\ell-1}^T$, and a weak approximation property $(A2)_α$ of Bramble type. Under these hypotheses, together with Richardson smoothing, the symmetric W-cycle and the variable V-cycle are known to be uniformly convergent, while the uniform convergence of the standard symmetric V-cycle has remained open. We answer this in the negative by two constructions. First, for every smoothing count $m$ we exhibit hierarchies of every depth satisfying (G3), Richardson admissibility with $C_R=1$, and $(A2)_α$ for every $α\in(0,1]$ with the sharp level-independent constant $C_{A2}^2=4m$, whose symmetric $V(m,m)$-cycle error operator has spectral radius $θ_m(1+2θ_m)>1$ already on three levels, where $θ_m=(1-\frac{1}{4m})^{2m}$, and growing geometrically with the depth; the family shows that any smoothing-count threshold $m_0$ that could restore uniform V-cycle convergence must grow at least quadratically in $C_{A2}$. Second, we prove that the same failure occurs in a completely standard discretisation: the cell-centred finite-volume hierarchy for a one-dimensional diffusion equation with a mesh-aligned coefficient jump $1:κ$ and harmonic (Samarskii) interface averaging satisfies (G3) exactly and $(A2)_{1/2}$ with a level-independent constant $C_{A2}=O(κ)$, yet for every $κ\ge3$ its symmetric $V(1,1)$-cycle with any admissible Richardson parameter, including the optimal one, diverges geometrically in the number of levels. In both constructions the W-cycle remains uniformly contractive, so the hypotheses separate the two cycles. All claims are verified numerically.

math.NA↗

arXiv2025

A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics

Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This approach reformulates the original equations on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM for a two-sided problem with transmission-type Robin boundary conditions. Our results show that, in the one dimensional space, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in $\varepsilon$. Furthermore, we provide numerical simulations that validate and illustrate the analytical result.

math.AP↗

arXiv2025

A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This reformulation extends the problem to a regular domain, incorporating boundary conditions via singular source terms. In this work, we analyze the convergence of a DDM approximation problem with transmission-type Neumann boundary conditions. We prove that the energy functional of the diffuse domain problem $Γ$--converges to the energy functional of the original problem as $\varepsilon \to 0$. Additionally, we show that the solution of the diffuse domain problem strongly converges in $H^1(Ω)$, up to a subsequence, to the solution of the original problem, as $\varepsilon \to 0$.

math.AP↗