[Paper Review] Two-parameter asymptotic expansions for elliptic equations with small geometric perturbation and high contrast ratio

Jingrun Chen, Ling Lin+2 more|arXiv (Cornell University)|Aug 15, 2017

Advanced Mathematical Modeling in Engineering10 references3 citations

TL;DR

This paper develops two-parameter asymptotic expansions for elliptic interface problems with small geometric perturbations and high contrast ratios in conductivity. By extending solutions to a fixed reference domain and applying Taylor expansion techniques, the authors derive asymptotic solutions of arbitrary order, revealing that Neumann or Robin boundary conditions emerge depending on the relative scaling of perturbation size and contrast ratio.

ABSTRACT

We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirich- let boundary condition and transmission condition, subject to the small geometric perturbation and the high contrast ratio of the conductivity. We consider two types of perturbations: the first corresponds to a thin layer coating a fixed bounded domain and the second is the per perturbation of the interface. As the perturbation size tends to zero and the ratio of the conductivities in two subdomains tends to zero, the two-parameter asymptotic expansions on the fixed reference domain are derived to any order after the single parameter expansions are solved be- forehand. Our main tool is the asymptotic analysis based on the Taylor expansions for the properly extended solutions on fixed domains. The Neumann boundary condition and Robin boundary condition arise in two-parameter expansions, depending on the relation of the geometric perturbation size and the contrast ratio.

Motivation & Objective

• To analyze the asymptotic behavior of solutions to elliptic interface problems under combined small geometric perturbations and high contrast ratios in conductivity.
• To derive two-parameter asymptotic expansions—accounting for both perturbation size ε and contrast ratio—on a fixed reference domain.
• To characterize the emergence of Neumann or Robin boundary conditions in the asymptotic expansions based on the relative scaling of ε and the contrast ratio.
• To provide a systematic framework for high-order asymptotic approximations in problems with thin layers or perturbed interfaces.
• To establish a rigorous analytical foundation for uncertainty quantification in problems with random geometric perturbations by reducing them to deterministic perturbation problems.

Proposed method

• Extend the solution $ u_ u $ defined on the perturbed domain $ D_ u $ to a fixed reference domain $ D $ using a diffeomorphic transformation based on the perturbation function $ h $.
• Apply Taylor expansions in the perturbation parameter $ \varepsilon $ to the extended solutions, enabling asymptotic expansion in powers of $ \varepsilon $.
• Derive a hierarchy of boundary value problems for each term in the two-parameter expansion by matching powers of $ \varepsilon $ and the contrast ratio.
• Solve the resulting equations recursively: zeroth-order terms satisfy Laplace or Poisson equations with interface and boundary conditions.
• Incorporate transmission conditions across the interface $ \Gamma $, with normal derivatives and tangential gradients appearing in higher-order terms.
• Use solvability conditions (e.g., integral constraints on fluxes) to uniquely determine constants in Neumann-type problems arising when $ \partial D^- \cap \partial D = \emptyset $.

Experimental results

Research questions

• RQ1 How do solutions to elliptic interface problems behave under simultaneous small geometric perturbations and high contrast ratios in conductivity?
• RQ2 What is the structure of the two-parameter asymptotic expansion when both the perturbation size $ \varepsilon $ and the contrast ratio are small?
• RQ3 Under what scaling regimes do Neumann or Robin-type boundary conditions emerge in the asymptotic model?
• RQ4 How can the solution be systematically expanded to arbitrary order using a fixed reference domain?
• RQ5 What role do the geometry of the interface and the smoothness of the perturbation function $ h $ play in the convergence and structure of the expansion?

Key findings

• The two-parameter asymptotic expansion is derived to arbitrary order by first solving the single-parameter expansion and then extending it via Taylor expansion of the extended solution.
• The leading-order term $ u_0^0 $ in the thin layer problem satisfies a harmonic equation in $ D^- $ with zero normal derivative on $ \Gamma $, and a Poisson equation in $ D^+ $ with Dirichlet data.
• The constant $ C_0 $ in the zeroth-order solution $ u_0^{-0} \equiv C_0 $ is uniquely determined by the solvability condition $ \int_\Gamma \partial_\mathbf{n} u_0^{0} \, dS_\Gamma = -\int_{D^-} f \, d\mathbf{x} $, which involves solving auxiliary harmonic and Poisson problems.
• For higher-order terms, the interface conditions involve both normal and tangential derivatives of lower-order solutions, with correction terms involving $ h $ and its derivatives.
• When $ \partial D^- \cap \partial D = \emptyset $, the Neumann problems for $ u_0^{-n} $ and $ u_1^{-n} $ are uniquely solvable only up to constants, which are fixed by integral solvability conditions.
• The method successfully captures the transition from Dirichlet to Neumann or Robin behavior in the asymptotic model depending on the relative scaling of $ \varepsilon $ and the contrast ratio.