Double Layer Potential Operator Research Articles

The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

For the Neumann-Poincaré (double layer potential) operator in the three-dimensional elasticity we establish asymptotic formulas for eigenvalues converging to the points of the essential spectrum and discuss geometric and mechanical meaning of coefficients in these formulas. In particular, we establish that for any body, there are infinitely many eigenvalues converging from above to each point of the essential spectrum. On the other hand, if there is a point where the boundary is concave (in particular, if the body contains cavities) then for each point of the essential spectrum there exists a sequence of eigenvalues converging to this point from below. The reasoning is based upon the representation of the Neumann-Poincaré operator as a zero order pseudodifferential operator on the boundary and the earlier results by the author on the eigenvalue asymptotics for polynomially compact pseudodifferential operators.

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm <1/2 as an operator on the natural trace space H^{1/2}(Gamma ) whenever Gamma is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H^{1/2}(Gamma ) for any sequence of finite-dimensional subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in H^{1/2}(Gamma ). Long-standing open questions are whether the essential norm is also <1/2 for D as an operator on L^2(Gamma ) for all Lipschitz Gamma in 2-d; or whether, for all Lipschitz Gamma in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators pm frac{1}{2}I+D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in L^2(Gamma ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ge 1/2, and examples with Lipschitz constant two for which the operators pm frac{1}{2}I +D are not coercive plus compact. We also give, for every C>0, examples of Lipschitz polyhedra for which the essential norm is ge C and for which lambda I+D is not a compact perturbation of a coercive operator for any real or complex lambda with |lambda |le C. We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the L^2(Gamma ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on C(Gamma ), equivalent to the standard supremum norm, for which the essential norm of D on C(Gamma ) is <1/2.

A Single-Layer Dual-Mesh Boundary Element Method for Multiscale Electromagnetic Modeling of Penetrable Objects in Layered Media

A surface integral representation of Maxwell's equations allows the efficient electromagnetic (EM) modeling of three-dimensional structures with a two-dimensional discretization, via the boundary element method (BEM). However, existing BEM formulations either lead to a poorly conditioned system matrix for multiscale problems, or are computationally expensive for objects embedded in layered substrates. This article presents a new BEM formulation which leverages the surface equivalence principle and Buffa-Christiansen basis functions defined on a dual mesh, to obtain a well-conditioned system matrix suitable for multiscale EM modeling. Unlike existing methods involving dual meshes, the proposed formulation avoids the double-layer potential operator for the surrounding medium, which may be a stratified substrate requiring the use of an advanced Green's function. This feature greatly alleviates the computational expense associated with the use of Buffa-Christiansen functions. Numerical examples drawn from several applications, including remote sensing, chip-level EM analysis, and metasurface modeling, demonstrate speed-ups ranging from 3x to 7x compared to state-of-the-art formulations.

SLIM: A Well-Conditioned Single-Source Boundary Element Method for Modeling Lossy Conductors in Layered Media

The boundary element method (BEM) enables the efficient electromagnetic modelling of lossy conductors with a surface-based discretization. Existing BEM techniques for conductor modelling require either expensive dual basis functions or the use of both single- and double-layer potential operators to obtain a well-conditioned system matrix. The associated computational cost is particularly significant when conductors are embedded in stratified media, and the expensive multilayer Green's function (MGF) must be invoked. In this work, a novel single-source BEM formulation is proposed, which leads to a well-conditioned system matrix without the need for dual basis functions. The proposed single-layer impedance matrix (SLIM) formulation does not require the double-layer potential to model the background medium, which reduces the cost associated with the MGF. The accuracy and efficiency of the proposed method is demonstrated through realistic examples drawn from different applications.

A global Tb Theorem for compactness and boundedness of Calderón-Zygmund operators

We prove a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on Lp(Rn), 1<p<∞. The result, whose proof does not require the property of accretivity, can be used to prove compactness of the Double Layer Potential operator on a wide class of domains.The study also provides conditions for boundedness of singular integral operators by means of non-accretive testing functions.

Boosting the Maxwell double layer potential using a right spin factor

We construct new spin singular integral equations for solving scattering problems for Maxwell’s equations, both against perfect conductors and in media with piecewise constant permittivity, permeability and conductivity, improving and extending earlier formulations by the author. These differ in a fundamental way from classical integral equations, which use double layer potential operators, and have the advantage of having a better condition number, in particular in Fredholm sense and on Lipschitz regular interfaces, and do not suffer from spurious resonances. The construction of the integral equations builds on the observation that the double layer potential factorises into a boundary value problem and an ansatz. We modify the ansatz, inspired by a non-selfadjoint local elliptic boundary condition for Dirac equations.

Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids

We use an approach based on pseudodifferential operators on Lie groupoids to study the double layer potentials on plane polygons. Let $$\Omega $$ be a simply connected polygon in $$\mathbb {R}^2$$ . Denote by K the double layer potential operator on $$\Omega $$ associated with the Laplace operator $$\Delta $$ . We show that the operator K belongs to the groupoid $$C^*$$ -algebra that the first named author has constructed in an earlier paper (Carvalho and Qiao in Cent Eur J Math 11(1):27–54, 2013). By combining this result with general results in groupoid $$C^*$$ -algebras, we prove that the operators $$\pm I + K$$ are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators $$\pm I + K$$ are invertible between suitable weighted Sobolev spaces through techniques from Mellin transform. The invertibility of these operators implies a solvability result in weighted Sobolev spaces for the interior and exterior Dirichlet problems on $$\Omega $$ .

Double layer potentials on three-dimensional wedges and pseudodifferential operators on Lie groupoids

Let W be a three-dimensional wedge, and K be the double layer potential operator associated to W and the Laplacian. We show that 12±K are isomorphisms between suitable weighted Sobolev spaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on W. Furthermore, we show that the double layer potential operator K is an element in C⁎(G)⊗M2(C), where G is the action (transformation) groupoid M⋊G, with G={(10ab):a∈R,b∈R+}, which is a Lie group, and M is a kind of compactification of G. This result can be used to prove the Fredholmness of 12+KΩ, where Ω is “a domain with edge singularities” and KΩ the double layer potential operator associated to the Laplacian and Ω.

Adaptive Wavelet BEM for Boundary Integral Equations: Theory and Numerical Experiments

ABSTRACTWe are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ℝ3. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution.

Тестирование возможностей открытого кода BEM++ по решению задач акустики

Testing of capabilities of open-source BEM++ code for simulation of acoustics problems at medium and high frequencies is presented. The BEM++ library is a universal tool, which allows to build discrete models for boundary integral operators (single-, double- and adjoint double-layer potential operators and hypersingular boundary operators) and solve boundary element method problems for Helmholtz, Laplace and Maxwell equations using Python libraries. Solution for the test problem of scattering plane wave on spherical obstacle with using BEM++ demonstrates good convergence with the results of analytical solutions. The relative errors satisfy to acceptable values 5% in solving engineering tasks, this fact allows to use this library as an alternative to commercial software. Capability of BEM++ library to calculate acoustic fields for frequencies from 5 Hz to 5 kHz enables move to solving more difficult engineering challenges of the aerospace industry. The main restriction for this is a time of computation, because only shared-memory technology of the code parallelization is implemented. However, open architecture of the library allows to remove this disadvantage. Meshes for BEM++ can have big size and be based on E geometric model with complex geometrical objects. Also, it should be noted, that for implementation to engineering practice it is desirable to integrate the library with existing interactive systems of pre- and post-processing, for example, with Salome.

Effective semi-analytic integration for hypersingular Galerkin boundary integral equations for the Helmholtz equation in 3D

We deal with the Galerkin discretization of the boundary integral equations corresponding to problems with the Helmholtz equation in 3D. Our main result is the semi-analytic integration for the bilinear form induced by the hypersingular operator. Such computations have already been proposed for the bilinear forms induced by the single-layer and the double-layer potential operators in the monograph The Fast Solution of Boundary Integral Equations by O. Steinbach and S. Rjasanow and we base our computations on these results.

Layer potentials beyond singular integral operators

We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A\in L_\infty, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.

Single and Double Layer Potentials on Domains with Conical Points I: Straight Cones

Let $${\Omega = \mathbb{R}^+ \omega}$$ be an open straight cone in $${\mathbb{R}^n, n\geq3}$$ , where $${\omega \subset S^{n-1}}$$ is a smooth subdomain of the unit sphere. Denote by K and S the double and single layer potential operators associated to Ω and the Laplace operator Δ. Let r be the distance to the origin. We consider a natural class of dilation invariant operators on ∂Ω, called Mellin convolution operators and show that $${K_a :=r^{a}Kr^{-a}}$$ and $${S_b := r^{b-\frac{1}{2}}Sr^{-b-\frac{1}{2}}}$$ are Mellin convolution operators for $${a \in (-1, n-1)}$$ and $${b \in (\frac{1}{2}, n-\frac{3}{2})}$$ . It is known that a Mellin convolution operator T is invertible if, and only if, its Mellin transform $${\hat T( \lambda)}$$ is invertible for any real λ. We establish a reduction procedure that relates the Mellin transforms of K a and S b to the single and, respectively, double layer potential operators associated to some other elliptic operators on ω, which can be shown to be invertible using the classical theory of layer potential operators on smooth domains. This reduction procedure thus allows us to prove that $${\frac{1}{2}\pm K}$$ and S are invertible between suitable weighted Sobolev spaces. A classical consequence of the invertibility of these operators is a solvability result in weighted Sobolev spaces for the Dirichlet problem on Ω.

Fast evaluation of singular BEM integrals based on tensor approximations

In this paper, we propose a method for the fast evaluation of integrals stemming from boundary element methods including discretisations of the classical single and double layer potential operators. Our method is based on the parametrisation of boundary elements in terms of a d-dimensional parameter tuple. We interpret the integral as a real-valued function f depending on d parameters and show that f is smooth in a d-dimensional box. A standard interpolation of f by polynomials leads to a d-dimensional tensor which is given by the values of f at the interpolation points. This tensor may be approximated in a low rank tensor format like the (CP) format or the $${{\mathcal {H}}}$$ -Tucker format. The tensor approximation has to be done only once and allows us to evaluate interpolants in $${{\mathcal{O}}(dr(m+1))}$$ operations in the (CP) format, or $${{\mathcal{O}}(dk^3+dk(m+1))}$$ operations in the $${{\mathcal H}}$$ -Tucker format, where m denotes the interpolation order and the ranks r, k are small integers. We demonstrate that highly accurate integral values can be obtained at very moderate costs.

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

A generalized eigenproblem for the Laplacian which arises in lightning

The following generalized eigenproblem is analyzed: Find u ∈ H 0 1 ( Ω ) , u ≠ 0 , and λ ∈ R such that 〈 ∇ u , ∇ v 〉 D = λ 〈 ∇ u , ∇ v 〉 Ω for all v ∈ H 0 1 ( Ω ) , where Ω ⊂ R n is a bounded domain, D is a subdomain with closure contained in Ω, and 〈 ⋅ , ⋅ 〉 Ω is the inner product 〈 ∇ u , ∇ v 〉 Ω = ∫ Ω ∇ u ⋅ ∇ v d x . It is proved that any f ∈ H 0 1 ( Ω ) can be expanded in terms of orthogonal eigenfunctions for the generalized eigenproblem. During the analysis, we present a new inner product on H 1 / 2 ( ∂ D ) with the following properties: (a) the norm associated with the inner product is equivalent to the usual norm on H 1 / 2 ( ∂ D ) , and (b) the double layer potential operator is self adjoint with respect to the new inner product and compact as a mapping from H 1 / 2 ( ∂ D ) into itself. The analysis identifies four classes of eigenfunctions for the generalized eigenproblem: 1. The function Π which is 1 on D and harmonic on Ω ∖ D ; the eigenvalue is 0. 2. Functions in H 0 1 ( Ω ) with support in Ω ∖ D ; the eigenvalue is 0. 3. Functions in H 0 1 ( Ω ) with support in D; the eigenvalue is 1. 4. Excluding Π, the harmonic extension of the eigenfunctions of a double layer potential on ∂ D. The eigenvalues are contained in the open interval ( 0 , 1 ) . The only possible accumulation point is λ = 1 / 2 . A positive lower bound for the smallest positive eigenvalue is obtained. These results can be used to evaluate the change in the electric potential due to a lightning discharge.

Interval analysis techniques for boundary value problems of elasticity in two dimensions

In this paper we prove that the L 2 spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in R 2 is within 10 −2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof.

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincare–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.

Spectral properties of parabolic layer potentials and transmission boundary problems in nonsmooth domains

We study the invertibility of $\lambda I+K$ in $L^p(\partial\Omega\times\mathbf{R})$, for $p$ near $2$ and $\lambda\in\mathbf{R}$, $|\lambda|\geq\sfrac12$, where $K$ is the caloric double layer potential operator and $\Omega$ is a Lipschitz domain. Applications to transmission boundary value problems are also presented.

Some counterexamples for the spectral-radius conjecture

The goal of this paper is to produce a series of counterexamples for the $L^p$ spectral radius conjecture, $1 <p <\infty$, for double-layer potential operators associated to a distinguished class of elliptic systems in polygonal domains in $\mathbb R^2$. More specifically the class under discussion is that of second-order elliptic systems in two dimensions whose coefficient tensor (with constant real entries) is symmetric and strictly positive definite. The general techniques employed are those of the Mellin transform and Calderón-Zygmund theory. For the case $p\in(1,4)$, we construct a computer-aided proof utilizing validated numerics based on interval analysis.