A Singularity Guided Nyström Method for Elastostatics on Two Dimensional Domains with Corners

We develop a comprehensive analytical and numerical framework for boundary integral equations (BIEs) of the 2D Lamé system on cornered domains. By applying local Mellin analysis on a wedge, we obtain a factorizable characteristic equation for the singular exponents of the boundary densities, and clarify their dependence on boundary conditions. The Fredholm well-posedness of the BIEs on cornered domains is proved in weighted Sobolev spaces. We further construct an explicit density-to-Taylor mapping for the BIE and show its invertibility for all but a countable set of angles. Based on these analytical results, we propose a singularity guided Nyström (SGN) scheme for the numerical solution of BIEs on cornered domains. The SGN uses the computed corner exponents and a Legendre-tail indicator to drive panel refinement. An error analysis that combines this refinement strategy with an exponentially accurate far-field quadrature rule is provided. Numerical experiments across various cornered geometries demonstrate that SGN obtains higher order accuracy than uniform Nyström method and reveal a crowding-limited regime for domains with re-entrant angles.

💡 Research Summary

The paper presents a comprehensive analytical and numerical framework for solving boundary integral equations (BIEs) of the two‑dimensional Lamé system on domains that contain corners. The authors begin by formulating the interior Dirichlet problem using the classical double‑layer potential representation, which leads to a second‑kind BIE of the form (-\frac{\theta}{2\pi}I + K). They then address two major challenges that arise on non‑smooth boundaries: (1) the Neumann‑Poincaré operator (K) is non‑compact and possesses a continuous spectrum, and (2) the unknown boundary density exhibits strong, often oscillatory, singularities near corners.

To overcome these difficulties, the paper adopts the Kondratiev‑Dauge weighted Sobolev space framework. By introducing a weight function based on the distance to each corner, the space (H^{s}_{\nu}(\Gamma)^2) captures the local singular behavior. The authors prove a Fredholm theorem (Theorem 2.2) stating that the BIE operator is Fredholm of index zero provided the shift (s-\nu) does not coincide with the real parts of the singular exponents or an integer. This result links the well‑posedness of the BIE directly to the corner singularities.

The core analytical contribution is a Mellin‑transform based corner analysis. By locally mapping a corner to a wedge and applying the Mellin transform in the radial variable, the BIE is reduced to a matrix family (A(z,\theta)) depending on a complex Mellin parameter (z) and the opening angle (\theta). The determinant of (A(z,\theta)) yields a factorizable characteristic equation whose zeros are the singular exponents ({z_{n,j}}). The equation splits into two branches, producing real power‑law exponents and complex exponents that generate logarithmic oscillations. The authors also construct an explicit density‑to‑Taylor mapping (B(\theta)) and prove its invertibility for all but a countable set of angles, thereby establishing a bridge between corner coefficients and smooth boundary data.

Guided by these analytical insights, the authors develop the Singulariy Guided Nyström (SGN) scheme. The algorithm proceeds as follows: (i) compute the singular exponents for each corner; (ii) use a multi‑exponent Legendre‑tail indicator to adaptively refine panels near corners, allocating more degrees of freedom where the exponents are larger; (iii) apply an exponentially accurate far‑field quadrature rule to handle interactions between well‑separated panels; (iv) assemble the Nyström discretization of the BIE and solve the resulting linear system. An a‑priori error analysis combines the local refinement error (behaving like (h^{p}) with (h) the smallest panel size) and the far‑field quadrature error (decaying like (e^{-cN}) with (N) quadrature points), showing that the total error can achieve high‑order convergence as long as the refinement and quadrature are balanced.

Numerical experiments cover several geometries: a single corner wedge, polygons with multiple corners, and domains containing re‑entrant angles (> π). In all cases SGN outperforms a uniform Nyström baseline, delivering at least a two‑fold increase in convergence rate. For re‑entrant corners the authors observe a “crowding‑limited” regime: the required panel density grows rapidly, eventually limiting attainable accuracy, a phenomenon predicted by the theory. Nevertheless, by imposing a practical panel‑size ceiling, SGN still attains high‑order accuracy.

In summary, the paper unifies Mellin symbol analysis, weighted Sobolev space theory, and singularity‑driven adaptive Nyström discretization into a coherent framework for 2D elastostatic BIEs on cornered domains. The results provide both rigorous well‑posedness guarantees and a practical high‑order numerical method, opening the way for extensions to three‑dimensional elasticity, heterogeneous materials, and time‑harmonic wave scattering.