Curvilinear Polygons Research Articles

Exponential convergence of hp FEM for spectral fractional diffusion in polygons

For the spectral fractional diffusion operator of order 2s, s in (0,1), in bounded, curvilinear polygonal domains varOmega subset {{mathbb {R}}}^2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm {mathbb {H}}^s(varOmega ). The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1-dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations invarOmega . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in varOmega , exponential convergence rates for solutions uin {mathbb {H}}^s(varOmega ) of mathcal {L}^s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towardspartial varOmega . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of mathcal {L}^{-s} combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations invarOmega . The present analysis for either approach extends to (polygonal subsets {widetilde{mathcal {M}}} of) analytic, compact 2-manifolds {mathcal {M}}, parametrized by a global, analytic chart chi with polygonal Euclidean parameter domain varOmega subset {{mathbb {R}}}^2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.

NURBS Enhanced Virtual Element Methods for the Spatial Discretization of the Multigroup Neutron Diffusion Equation on Curvilinear Polygonal Meshes

The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretization methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. Recently, a new formulation of CG-VEM was introduced which can construct VEM spaces on polygons with curvilinear edges. This paper presents the application of the curved VEM to the multigroup neutron diffusion equation and demonstrates its benefits over the conventional straight-sided VEM for a number of benchmark verification test cases with curvilinear domains. These domains were constructed using a topological data-structure developed as part of this paper, based on the doubly-connected edge list, with curves and surfaces both represented using non-uniform rational B-splines (NURBS). This data-structure is used both to specify the geometry of the reactor and to represent the curvilinear polygonal mesh. We also present two separate methods of performing integrations on curvilinear polygons, one for homogeneous functions and one for non-homogeneous functions.

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.

Quadrature for implicitly-defined finite element functions on curvilinear polygons

H1-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined as solutions of Poisson problems having polynomial source and boundary data. Recently, such methods have been extended to allow for mesh cells that are curvilinear polygons. Such extensions present new challenges for determining suitable quadratures. We describe an approach for integrating products of these implicitly defined functions, as well as products of their gradients, that reduces integrals on cells to integrals along their boundaries. Numerical experiments illustrate the practical performance of the proposed methods.

Effective operators for Robin eigenvalues in domains with corners

We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schr\"odinger-type operator on the boundary of the domain with boundary conditions at the corners.

Inverse Steklov Spectral Problem for Curvilinear Polygons

Abstract This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $\pi $, we prove that the asymptotics of Steklov eigenvalues obtained in [ 20] determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other techniques, we use a version of the Hadamard–Weierstrass factorization theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots.

Trefftz Finite Elements on Curvilinear Polygons

We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be “polynomial” of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local finite element spaces, and discuss how to work with these edge polynomial spaces in practice. An interpolation operator is introduced for the resulting finite elements, and we prove that it provides optimal order convergence for interpolation error under reasonable assumptions. We provide a description of the integral equation approach used for the examples in this paper, which was recently developed precisely with these applications in mind. A few numerical examples illustrate this optimal order convergence of the finite element solution on some families of meshes in which every element has at least one curved edge. We also demonstrate that it is possible to exploit the approximation power of locally singular functions that may exist in our finite element spaces in order to achieve optimal order convergence without the typical adaptive refinement toward singular points.

Distributed and boundary expressions of first and second order shape derivatives in nonsmooth domains

We study distributed and boundary integral expressions of Eulerian and Fréchet shape derivatives for several classes of nonsmooth domains such as open sets, Lipschitz domains, polygons and curvilinear polygons, semiconvex and convex domains. For general shape functionals, we establish relations between distributed Eulerian and Fréchet shape derivatives in tensor form for Lipschitz domains, and infer two types of boundary expressions for Lipschitz and C1-domains. We then focus on the particular case of the Dirichlet energy, for which we compute first and second order distributed shape derivatives in tensor form. Depending on the type of nonsmooth domain, different boundary expressions can be derived from the distributed expressions. This requires a careful study of the regularity of the solution to the Dirichlet Laplacian in nonsmooth domains. These results are applied to obtain a matricial expression of the second order shape derivative for polygons.

Characterization of the essential spectrum of the Neumann–Poincaré operator in 2D domains with corner via Weyl sequences

The Neumann–Poincaré (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely related to the well-posedness of these PDE’s, in the typical case where one considers a bounded inclusion of homogeneous plasmonic metamaterial embedded in a homogeneous background dielectric medium. In a recent work, M. Perfekt and M. Putinar have shown that the NP operator of a 2D curvilinear polygon has an essential spectrum, which depends only on the angles of the corners. Their proof is based on quasi-conformal mappings and techniques from complex-analysis. In this work, we characterise the spectrum of the NP operator for a 2D domain with corners in terms of elliptic corner singularity functions, which gives insight on the behaviour of generalized eigenmodes.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

A Nyström-based finite element method on polygonal elements

We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via Nyström discretizations of associated integral equations, allowing for curvilinear polygons and non-polynomial boundary data. Several experiments demonstrate the approximation quality of interpolated functions in these spaces.

Spectral asymptotics for Robin Laplacians on polygonal domains

Let Ω be a curvilinear polygon and QΩγ be the Laplacian in L2(Ω), QΩγψ=−Δψ, with the Robin boundary condition ∂νψ=γψ, where ∂ν is the outer normal derivative and γ>0. We are interested in the behavior of the eigenvalues of QΩγ as γ becomes large. We prove that the asymptotics of the first eigenvalues of QγΩ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with ∂Ω. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of QΩγ for a threshold depending on γ, and show that the leading term is the same as for smooth domains.

Invertibility Properties of Singular Integral Operators Associated with the Lamé and Stokes Systems on Infinite Sectors in Two Dimensions

In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on \(L^p(\partial \Omega )\), \(1<p<\infty \), whenever \(\Omega \) is an infinite sector in \({\mathbb {R}}^2\). This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lame system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector \(\Omega \) and the differential operator in question.

Numerical solution of boundary integral equations of the plane theory of elasticity in curvilinear polygons

A numerical method for solving boundary integral equations of the plane theory of elasticity in domains with piecewise analytic boundaries and a finite number of corner points is proposed. This method is based on the application of a family of composite quadrature formulas on condensing grids. It is proved that the proposed method is exponentially convergent with respect to the number of quadrature nodes in use.

2D Quaternionic Time-Harmonic Maxwell System in Elliptic Coordinates

In this paper we consider the 2D time–harmonic Maxwell equations in elliptic coordinates through certain quaternionic perturbed Dirac operator. The main goal is aimed to analyze an electromagnetic Dirichlet problem for a curvilinear polygon with rectifiable boundary in \({\mathbb{R}^2}\). In addition, we provide an integral representation formula for electromagnetic fields that resembles the classical Stratton-Chu formula. The importance of the problem for applications makes it worthy of consideration.

Numerical solution of boundary integral equations on curvilinear polygons

An approximate method of solving an integral equation of the potential theory for a Dirichlet problem for the Laplace operator is proposed in the case when domains are curvilinear polygons with piecewise analytic boundaries. The proposed method is exponentially convergent with respect to the number of quadrature nodes in use.

On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions

By Birman and Skvortsov it is known that if Ω is a planar curvilinear polygon with n non-convex corners then the Laplace operator with domain H2(Ω)∩H01(Ω) is a closed symmetric operator with deficiency indices (n,n). Here we provide a Kreĭn-type resolvent formula for any self-adjoint extensions of such an operator, i.e. for the set of self-adjoint non-Friedrichs Dirichlet Laplacians on Ω, and show that any element in this set is the norm resolvent limit of a suitable sequence of Friedrichs–Dirichlet Laplacians with n point interactions.

An algebraic cubature formula on curvilinear polygons

Abstract We implement in Matlab a Gauss-like cubature formula on bivariate domains whose boundary is a piecewise smooth Jordan curve (curvilinear polygons). The key tools are Green’s integral formula, together with the recent software package Chebfun to approximate the boundary curve close to machine precision by piecewise Chebyshev interpolation. Several tests are presented, including some comparisons of this new routine ChebfunGauss with the recent SplineGauss that approximates the boundary by splines.

The inradius, the first eigenvalue, and the torsional rigidity of curvilinear polygons

Let λ1, P, and ρ denote the first eigenvalue of the Dirichlet Laplacian, the torsional rigidity, and the inradius of a planar domain Ω, respectively. In this paper, we prove several inequalities for λ1, P, and ρ in the case when Ω is a curvilinear polygon with n sides, each of which is a smooth arc of curvature at most κ. Our main proofs rely on the method of dissymmetrization and on a special geometrical ‘containment theorem’ for curvilinear polygons. For rectilinear n-gons, which constitute a proper subclass of curvilinear n-gons with curvature at most 0, these inequalities were established by the first author in 1992. In the simplest particular case of Euclidean triangles T, the inequality linking the first eigenvalue and the inradius of T is equivalent to the inequality λ 1 ⩽ π2 L2/9 A2, where A and L denote the area and perimeter of T, respectively, which was recently discussed by Freitas (Proc. Amer. Math. Soc. 134 (2006) 2083–2089) and Siudeja (Michigan Math. J. 55 (2007) 243–254).

High frequency scattering by convex curvilinear polygons

We consider the scattering of a time-harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave. Here we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes; a uniform mesh on illuminated sides, and graded meshes refined towards the corners of the polygon on illuminated and shadow sides. Numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases.