arXiv subjects

Tristan Goodwill

Publications and source records attributed to Tristan Goodwill.

11 recordsLinked to original sources

arXiv2026

The three-dimensional Neumann Green's function for general surfaces: singular asymptotics and boundary integral methods

We present an asymptotic analysis and high-order boundary integral method for the three-dimensional Neumann Green's function in general closed and smooth geometries. The Neumann Green's function is a fundamental quantity which arises in numerous fields of science and engineering. In the application of singular perturbation methods to strongly localized reactions and diffusive transport, the Green's function plays the key role in mediating global dynamics. However, this essential quantity can only be determined in closed-form for a limited set of geometries. The Green's function for the Laplacian is an elliptic problem with a Dirac forcing term. Accurate resolution of the solution requires a careful decomposition into a singular and a regular part. The bulk scenario is where the source is placed off surface and the singularity is given by the free-space function. In the surface case, where the source is placed at a curved point on the boundary, we use asymptotic analysis to determine a three-term singularity structure. With explicit knowledge of these singularities, we develop a high-order boundary integral method for the determination of the remaining regular part. To resolve the singular boundary data, our integral method uses a custom discretization with Duffy patches near the source. We validate our method using several test cases in which closed-form solutions can be developed, including spheres, prolate spheroids and constructed domains. We demonstrate the applicability of our method to address some open problems in narrow capture theory.

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arXiv2026

Integral equations for flexural scattering problems with periodic boundaries

We develop a method for computing the scattering of flexural waves off of a periodic wall or a periodic line of scatterers. These waves model the fluctuations of thin plates with periodic clamped, supported, or free edges. We use the Floquet-Bloch transform to convert the problem into a collection of uncoupled quasi-periodic problems. We then solve each quasi-periodic problem efficiently and accurately using a novel integral equation based on the quasi-periodic flexural Green's function. Finally, we show how the proposed method can be used to simulate scattering from junctions of semi-infinite lines of scatterers.

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arXiv2026

A parametrix for the surface Stokes equation

We introduce an integral equation formulation of the surface Stokes equations, constructed using two-dimensional Stokeslets. The resulting integral equations are Fredholm integral equations of the second kind and can be discretized to high order using standard tools. Since the resulting discrete linear systems are dense, we describe and analyze a proxy shell method to construct fast direct solvers for these systems. The properties of our integral equation, and the performance of the resulting numerical scheme, are illustrated with several representative numerical examples.

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arXiv2025

Surface layers and linearized water waves: a boundary integral equation framework

The dynamics of surface waves traveling along the boundary of a liquid medium are changed by the presence of floating plates and membranes, contributing to a number of important phenomena in a wide range of applications. Mathematically, if the fluid is only partly covered by a plate or membrane, the order of derivatives of the surface-boundary conditions jump between regions of the surface. In this work, we consider a general class of problems for infinite depth linearized surface waves in which the plate or membrane has a compact hole or multiple holes. For this class of problems, we describe a general integral equation approach, and for two important examples, the partial membrane and the polynya, we analyze the resulting boundary integral equations. In particular, we show that they are Fredholm second kind and discuss key properties of their solutions. We develop flexible and fast algorithms for discretizing and solving these equations, and demonstrate their robustness and scalability in resolving surface wave phenomena through several numerical examples.

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arXiv2025

A domain decomposition method for computing the scattering matrix of waveguide circuits

We analyze and develop numerical methods for time-harmonic wave scattering in metallic waveguide structures of infinite extent. We show that radiation boundary conditions formulated via projectors onto outgoing modes determine the coefficients of propagating modes uniquely, even when the structure supports trapped modes. Building on this, we introduce a fast divide-and-conquer solver that constructs solution operators on subdomains as impedance-to-impedance maps and couples them by enforcing continuity conditions across their interfaces. For Dirichlet waveguides, the computation of impedance-to-impedance maps requires the solution of mixed Dirichlet-Impedance boundary value problems. We construct a second-kind Fredholm integral equation that avoids near-hypersingular operators, requiring only integral operators whose kernels are at most weakly singular. Numerical experiments on large structures with many circuit elements demonstrate substantial efficiency gains: the proposed approach typically outperforms state-of-the-art fast iterative and fast direct solvers by one to two orders of magnitude.

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arXiv2025

Fast Multipole Method with Complex Coordinates

In this work we present a variant of the fast multipole method (FMM) for efficiently evaluating standard layer potentials on geometries with complex coordinates in two and three dimensions. The complex scaled boundary integral method for the efficient solution of scattering problems on unbounded domains results in complex point locations upon discretization. Classical real-coordinate FMMs are no longer applicable, hindering the use of this approach for large-scale problems. Here we develop the complex-coordinate FMM based on the analytic continuation of certain special function identities used in the construction of the classical FMM. To achieve the same linear time complexity as the classical FMM, we construct a hierarchical tree based solely on the real parts of the complex point locations, and derive convergence rates for truncated expansions when the imaginary parts of the locations are a Lipschitz function of the corresponding real parts. We demonstrate the efficiency of our approach through several numerical examples and illustrate its application for solving large-scale time-harmonic water wave problems and Helmholtz transmission problems.

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arXiv2025

Complex Scaling for the Junction of Semi-infinite Gratings

We present and analyze an integral equation method for the scattering of a non-periodic source from a geometry consisting of two semi-infinite, periodic structures glued together in two dimensions. The two structures may involve a periodic wall, several layers of transmission surfaces with a shared period, or periodic sets of obstacles. This integral equation is posed on the infinite interface between the two periodic structures using kernels built out of the Green's function for each structure. To combat the slow decay of the Green's function, we also show that our integral equation can be analytically continued into the complex plane, where it can be truncated with exponential accuracy. A careful analysis of the domain Green's functions far from the periodic structure is then used to prove that the analytically continued equation is Fredholm index zero. Finally, we show that the solution we generate satisfies a radiation condition and demonstrate an efficient and high order solver for this problem.

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arXiv2025

Complex scaling for open waveguides

In this work we analyze the complex scaling method applied to the problem of time-harmonic scalar wave propagation in junctions between `leaky,' or open dielectric waveguides. In [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674, arXiv:2411.11204], it was shown that under suitable assumptions the problem can be reduced to a system of Fredholm second-kind integral equations on an infinite interface, transverse to the waveguides. Here, we show that the kernels appearing in the integral equation admit a rapidly decaying analytic continuation on certain natural totally real submanifolds of $\mathbb{C}^2.$ We then show that for suitable, physically-meaningful, boundary data the resulting solutions to the integral equations themselves admit analytic continuation and satisfy related asymptotic estimates. By deforming the integral equation to a suitable contour, the decay in the kernels, density, and data enable straightforward discretization and truncation, with an error that decays exponentially in the truncation length. We illustrate our results with several representative numerical examples.

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arXiv2024

A numerical method for scattering problems with unbounded interfaces

We introduce a new class of computationally tractable scattering problems in unbounded domains, which we call decomposable problems. In these decomposable problems, the computational domain can be split into a finite collection of subdomains in which the scatterer has a "simple" structure. A subdomain is simple if the domain Green's function for this subdomain is either available analytically or can be computed numerically with arbitrary accuracy by a tractable method. These domain Green's functions are then used to reformulate the scattering problem as a system of boundary integral equations on the union of the subdomain boundaries. This reformulation gives a practical numerical method, as the resulting integral equations can then be solved, to any desired degree of accuracy, by using coordinate complexification over a finite interval, and standard discretization techniques.

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arXiv2024

A parametrix method for elliptic surface PDEs

Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries.

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arXiv2021

An interface formulation of the Laplace-Beltrami problem on piecewise smooth surfaces

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). In particular, the Hodge decomposition of vector fields tangent to a surface can be computed by solving a sequence of Laplace-Beltrami problems. Such decompositions are very important in magnetostatic calculations and in various plasma and fluid flow problems. In this work we develop $L^2$-invertibility theory for the Laplace-Beltrami operator on piecewise smooth surfaces, extending earlier weak formulations and integral equation approaches on smooth surfaces. Furthermore, we reformulate the weak form of the problem as an interface problem with continuity conditions across edges of adjacent piecewise smooth panels of the surface. We then provide high-order numerical examples along surfaces of revolution to support our analysis, and discuss numerical extensions to general surfaces embedded in three dimensions.

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