Well-posed PDE and integral equation formulations for scattering by fractal screens

We consider time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens embedded in $\R^n$ for $n=2$ or $3$. In contrast to previous studies in which the screen is assumed to be a bounded Lipschitz (or smoother) relatively open subset of the plane, we consider screens occupying arbitrary bounded subsets. Thus our study includes cases where the screen is a relatively open set with a fractal boundary, and cases where the screen is fractal with empty interior. We elucidate for which screen geometries the classical formulations of screen scattering are well-posed, showing that the classical formulation for sound-hard scattering is not well-posed if the screen boundary has Hausdorff dimension greater than $n-2$. Our main contribution is to propose novel well-posed boundary integral equation and boundary value problem formulations, valid for arbitrary bounded screens. In fact, we show that for sufficiently irregular screens there exist whole families of well-posed formulations, with infinitely many distinct solutions, the distinct formulations distinguished by the sense in which the boundary conditions are understood. To select the physically correct solution we propose limiting geometry principles, taking the limit of solutions for a sequence of more regular screens converging to the screen we are interested in, this a natural procedure for those fractal screens for which there exists a standard sequence of prefractal approximations. We present examples exhibiting interesting physical behaviours, including penetration of waves through screens with "holes" in them, where the "holes" have no interior points, so that the screen and its closure scatter differently. Our results depend on subtle and interesting properties of fractional Sobolev spaces on non-Lipschitz sets

Acoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximation

We study time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a planar screen (a "crack" in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset $\Gamma$ of the hyperplane $\mathbb{R}^{n-1}\times \{0\}$, on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, $\Gamma$ can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such $\Gamma$ we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when $\partial\Gamma$ is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother "prefractal" screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake

Integral equation methods for acoustic scattering by fractals

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ is uniformly of $d$-dimensional Hausdorff dimension in a sense we make precise (a $d$-set), the operator in our equation is an integral operator on $\Gamma$ with respect to $d$-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code

Boundary element methods for acoustic scattering by fractal screens

We study boundary element methods for time-harmonic scattering in R^n (n=2,3) by a fractal planar screen, assumed to be a non-empty bounded subset Gamma of the hyperplane Gamma_\infty=R^{n-1}\times \{0\}. We consider two distinct cases: (i) Gamma is a relatively open subset of Gamma_\infty with fractal boundary (e.g. the interior of the Koch snowflake in the case n=3); (ii) Gamma is a compact fractal subset of Gamma_\infty with empty interior (e.g. the Sierpinski triangle in the case n=3). In both cases our numerical simulation strategy involves approximating the fractal screen Gamma by a sequence of smoother "prefractal" screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations. We prove sufficient conditions on the mesh sizes guaranteeing convergence to the limiting fractal solution, using the framework of Mosco convergence. We also provide numerical examples illustrating our theoretical results

On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

This paper concerns the following question: given a subset E of Rn with empty interior and an integrability parameter 1<p<infinity, what is the maximal regularity s in R for which there exists a non-zero distribution in the Bessel potential Sobolev space Hs,p(Rn) that is supported in E? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of E, sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of p, together with the sets of values of p for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterising the regularity that can be achieved on certain special classes of sets, such as d-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Accelerated Calderón Preconditioning for Electromagnetic Scattering by Multiple Absorbing Dielectric Objects

We consider electromagnetic scattering by multiple absorbing dielectric objects using the PMCHWT boundary integral equation formulation. Galerkin discretisation of this formulation leads to ill-conditioned linear systems, and Calderón preconditioning, an operator-based approach, can be used to remedy this. To obtain a stable discretisation of the operator products that arise in this approach, the use of a dual mesh defined on a barycentrically refined grid needs to be considered, increasing memory consumption. Furthermore, to capture the oscillatory solution of the electromagnetic waves, the mesh needs to be refined with respect to frequency, making the simulation of high-frequency problems very expensive. This thesis presents two complementary approaches to minimising memory cost and computation time (for assembly and solution): modification of the preconditioning operator, and a bi-parametric implementation. The former aims to minimise the number of operators used in the preconditioner to reduce the additional matrix-vector products performed, and the memory cost, while still maintaining a sufficient preconditioning effect. The latter uses two distinct sets of parameters during assembly, to minimise assembly and solution time as well as memory. The operator is assembled with a more expensive set of parameters to obtain an accurate solution. The preconditioner, which is discretised using the expensive dual basis functions, is assembled with a cheaper set of parameters. The two approaches are explained in the context of a series of model problems, then applied to realistic ice crystal configurations found in cirrus clouds. They are shown to deliver a reduction of 99% in memory cost and at least 80% in computation time, for the highest frequency considered. The accelerated formulations have been used at the Met Office to create a new database of the scattering properties of atmospheric ice crystals for future numerical weather prediction. A brief description of that work is also presented in the thesis