Hypersingular Operator Research Articles

A p‐adaptive algorithm for the BEM with the hypersingular operator on the plane screen

AbstractWe propose a p‐adaptive algorithm for the Galerkin method solving the hypersingular integral operator of the Laplacian on the plane screen. The error indicators/estimators are based on projections of the actual error onto local subspaces. These subspaces are defined by decompositions of specially designed enriched ansatz spaces. Our algorithm uses different strategies for the refinement and the stopping criterion. The error estimator that stops the algorithm is based on an overlapping decomposition of an ansatz space that is defined by mesh refinement. The error indicators that steer the p‐refinement are computed via an almost direct decomposition of an enriched ansatz space that uses the same mesh but higher polynomial degrees. Numerical results support the efficiency of our algorithm. Copyright © 2001 John Wiley & Sons, Ltd.

On certain one- and two-dimensional hypersingular integral equations

For two-dimensional singular and hypersingular integrals more general definitions than the traditional ones are introduced. For a hypersingular operator on a sphere a new spectral relation is obtained. Quadrature formulae of the kind of discrete vortex pairs for one-dimensional and of the kind of closed vortex frames for two-dimensional hypersingular integrals are considered; questions on their convergence are discussed, as well as the convergence of numerical solutions to the corresponding hypersingular integral equations on a finite line interval and a circle. An experiment on the numerical solution of a hypersingular integral equation on a sphere is carried out, which demonstrates analogies between numerical solutions of hypersingular integral equations on a finite interval and a sphere.

A scalar boundary integrodifferential equation for eddy current problems using an impedance boundary condition

At low frequencies the time harmonic electromagnetic fields exterior to a lossy, highly conducting and possibly magnetic body can be described by the eddy current approximation of Maxwell’s equations with impedance or Leontovich [18] boundary conditions if the so called penetration depth is small. We show how to reduce this problem to a scalarr boundary integrodifferential equation on the surface Γ of the conductor. Strong ellipticity of the associated nonsymmetric, hypersingular operator is established. Convergence of O(h ) of the Ohmic losses for piecewise linear, continuous boundary elements is established theoretically and numerically.

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincare and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

An additive Schwarz preconditioner for p -version boundary element approximation of the hypersingular operator in three dimensions

Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned system is bounded by \((1+\log p)^2\) for an open surface and \((1+\log p)\) for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results in a further degradation of the condition number by a factor \(p(1+\log p)\).

On boundary integral operators for the Laplace and the Helmholtz equations and their discretisations

Boundary integral operators for the solutions of the Laplace and the Helmholtz equations are considered. These are classical strongly elliptic pseudodifferential operators of integer orders α, mapping the Sobolev space H r to H { r-α} . We study the spectral properties of the single layer Laplacian potential operator and its tangential derivative, and also the double layer Laplacian potential operator and its normal derivative, the hypersingular operator, over a circle boundary. We extend the analysis to the Helmholtz potential operators. We derive important analytical results for the elements of the discrete operators and their eigenvalues and eigenvectors.

An iterative substructuring method for the $p$ -version of the boundary element method for hypersingular integral operators in three dimensions

We study preconditioners for the \(p\)-version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by \(C(1+\log p)\). Here, \(p\) is the polynomial degree of the ansatz functions and \(C\) is a constant which is independent of \(p\) and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported.

Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation

Discretization of boundary integral equations leads, in general, to fully populated complex valued non-Hermitian systems of equations. In this paper we consider the efficient solution of these boundary element systems by preconditioned iterative methods of Krylov subspace type. We devise preconditioners based on the splitting of the boundary integral operators into smooth and non-smooth parts and show these to be extremely efficient. The methods are applied to the boundary element solution of the Burton and Miller formulation of the exterior Helmholtz problem which includes the derivative of the double layer Helmholtz potential—a hypersingular operator. © 1998 John Wiley & Sons, Ltd.

Multigrid Schur Complement Preconditioning in BE‐DD Methods

AbstractWe consider Domain Decomposition (DD) Boundary Element (BE) equations and analyze a multigrid preconditioner, derived from the hypersingular operator, for the Schur complement. The robustness of the preconditioner with respect to the number of subdomains is being investigated.

Adaptive Boundary Element Mevthods for Some First Kind Integral Equations

In this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the Dirichlet problem and the Neumann problem for the Laplacian in a two-dimensional Lipschitz domain. For the h-version of the finite element Galerkin discretization of the single layer potential and the hypersingular operator, we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm (up to a multiplicative constant). Following Eriksson and Johnson this yields adaptive algorithms steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate.

Lp-bounds for hypersingular integral operators along curves

<i>L</i>^<i>p</i>-bounds for hypersingular integral operators along curves

A posterior error estimates for hp—boundary element methods

This paper presents a posterior error estimates for the hp-version of the boundary element method. We dicuss two first kund integral operator equations, namely Symm's integral equation and the equation with a hypersingular operator. The computable upper error bounds indicate an algorithm for the automatic hp-adaptive mesh-refnement. The efficiency of this method is shown by numerical experiments yielding almost optimal convergence even in the presence of corner singularities.

A Posteriori Error Estimates for Boundary Element Methods

This paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples, namely Symm's integral equation, an integral equation with a hypersingular operator, and a boundary integral equation for a transmission problem. Based on these estimates, an analog of Eriksson and Johnson's adaptive finite element method is proposed for the h -version of the Galerkin boundary element method for integral equations of the first kind. The efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities. The construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations, and we refer to the pioneering work of Babuska and Miller (3) and Eriksson and Johnson (10, 11). Whereas the main features ofadaptivity for finite element methods now seem to be visible, and the door is open to implementation (15), comparably little is known for boundary element methods for integral equations (see e.g. (1, 13, 18, 19,24)). In this paper a new adaptive «-version of the Galerkin discretization for the boundary element method is presented based on a posteriori error estimates. A general framework for these a posteriori error estimates is derived in §2, and three examples are discussed in §§3-5 involving the Dirichlet problem, the Neumann problem (for a closed and an open surface), and a transmission problem for the Laplacian, leading to integral equations with strongly elliptic pseudodifferential operators. Even for smooth data the lack of regularity of the solution near corners (of a polygonal domain Q) leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used. In practical problems such information is missing, e.g., when we have singular (or nearly singular) data and the main problem is how to balance a graded mesh refinement towards singularities and a global

Integral Equation Method via Domain Decomposition and Collocation for Scattering Problems

In this paper an exterior domain decomposition (DD) method based on the boundary element (BE) formulation for the solutions of two or three-dimensional time-harmonic scattering problems in acoustic media is described. It is known that the requirement of large memory and intensive computation has been one of the major obstacles for solving large scale high-frequency acoustic systems using the traditional nonlocal BE formulations due to the fully populated resultant matrix generated from the BE discretization. The essence of this study is to decouple, through DD of the problem-defined exterior region, the original problem into arbitrary subproblems with data sharing only at the interfaces. By decomposing the exterior infinite domain into appropriate number of infinite subdomains, this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized, blockwise-banded system of discretized equations, for which the solution requires only O(N) multiplications, where N is the number of unknowns on the scatterer surface. The size of an individual submatrix that is associated with a subdomain may be determined by the user, and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated. In addition, the method may suit for parallel processing since the data associated with each subdomain (impedance matrices, load vectors, etc.) may be generated in parallel, and the communication needed will be only for the interface values. Most significantly, unlike the existing boundary integral-based formulations valid for all frequencies, our method avoids the use of both the hypersingular operators and the double integrals, therefore reducing the computational effort. Numerical experiments have been conducted for rigid cylindrical scatterers subjected to a plane incident wave. The results have demonstrated the accuracy of the method for wave numbers ranging from 0 to 30, both directly on the scatterer and in the far-field, and have confirmed that the procedure is valid for critical frequencies.

A posteriori error estimates for boundary element methods

This paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples, namely Symm’s integral equation, an integral equation with a hypersingular operator, and a boundary integral equation for a transmission problem. Based on these estimates, an analog of Eriksson and Johnson’s adaptive finite element method is proposed for the h-version of the Galerkin boundary element method for integral equations of the first kind. The efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities.

A solution method for hypersingular integral equations

The Neumann problem for the Helmholtz equation is considered. The double-layer potential is used to reduce the problem to a hypersingular integral equation. The properties of the hypersingular operator in a neighborhood lead to a method for approximate solution of the hypersingular equation with an arbitrary contour. Some numerical results are reported.

A coupled FE and boundary integral equation method based on exterior domain decomposition for fluid-structure interface problems

A coupled finite element and exterior domain decomposition-based boundary integral formulations for the solutions of two- or three-dimensional time-harmonic fluid-structure interaction problems is described in this paper. It is known that the memory limitation of computers has been one of the major obstacles for solving large scale high frequency fluid-structure interface systems using various existing nonlocal finite element and boundary integral equation coupling techniques due to the fully populated resultant matrix generated from the boundary integral equation representation. The essence of this study is to decompose, through domain decomposition of the exterior region, the original exterior problem into arbitrary subproblems with data sharing only at the interfaces. By decomposing the exterior infinite domain into an appropriate number of infinite subdomains, this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized, blockwise-banded system of discretized equations. The size of an individual submatrix (i.e. a block) that is associated with an exterior subdomain may be decided by the user, and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated. In addition, the method is suited for parallel processing since the data associated with each subdomain (impedance matrices, load vectors, etc.) may be generated in parallel, and the communication needed will be only for the interface values. Most significantly, unlike the existing coupled finite element and boundary integral equation techniques that are valid for all frequencies, our method avoids the use of both the hypersingular operator and the double integrals, therefore reducing the computational complexity. Numerical experiments have been performed for elastic cylindrical shells subjected to a plane incident wave. The results have demonstrated the accuracy of the method for wavenumbers ranging from 0 to 30, both directly on the shell and in the far field, and have confirmed that the procedure is valid for all frequencies.

Exterior stable domain segmentation integral equation method for scattering problems

AbstractThis paper is concerned with the development of an exterior domain segmentation method for the solution of two‐ or three‐dimensional time‐harmonic scattering problems in acoustic media. This method, based on a variational localized, symmetric, boundary integral equation formulation leads, upon discretization, to a sparse system of algebraic equations whose solution requires only O(N) multiplications, where N is the number of unknown nodal pressures on the scatterer surface. The new procedure is analogous to the one developed recently1 except that in the present formulation we avoid completely the use of the hypersingular operator, thereby reducing the computational complexity. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers, ka, ranging from 0 to 30, both directly on the scatterer and in the far field, and to confirm that, contrary to standard boundary integral equation formulations, the present procedure is valid for critical frequencies.

Hypersingular integral operators in diffraction problems of electromagnetic waves on open surfaces

The three-dimensional problem of diffraction of a monochromatic electromagnetic field on an open ideally conducting surface has been reduced in [1] to the solution of an integrodifferential equation for the induced density of surface currents. In [2] this equation has been represented in hypersingular form, with the divergent integral understood in the sense of Hadamard finite part. In the present paper, we examine the properties of hypersingular operators and prove boundedness of the direct operator in the Banach spaces of tangential fields with a H61der-continuous surface divergence. An algorithm for numerical solution of the hypersingular equation is proposed, based on approximation of the operator by a basis system of piecewise-linear functions. Consider the mathematical formulation of the problem of diffraction of electromagnetic waves on an ideally conducting open surface S E C 2,a with the edge C O E C 2,a, which is embedded in an infinite homogeneous isotropic medium with dielectric permittivity e, magnetic permeability /z, and conductivity a. The electromagnetic wave with harmonic time dependence and frequency c~ is described by the electric and magnetic field vectors

Unified symmetric finite element and boundary integral variational coupling methods for linear fluid–structure interaction

AbstractThis article concerns the development of energy‐based variational formulations and their corresponding finite element–boundary element Rayleigh–Ritz approximations for solving the time‐harmonic vibration and scattering problem of an inhomogeneous penetrable fluid or solid object immersed in a compressible, inviscid, homogeneous fluid. The resulting coupled finite element and boundary integral methods (FEM‐BEM) have the following attractive features: (1) Separate direct and complementary variational principles lead naturally to several alternative structure variable and fluid variable methodologies. (2) The solution in the exterior region is represented by a combined single‐ and double‐layer potential which ensures the validity of the methods for all wave numbers; even though this representation introduces hypersingular integrals, for actual computations the hypersingular operator may be rewritten in terms of single‐layer potentials, which can be integrated by standard techniques. (3) Since the discretized equations for the interior region and for the boundary are derived from the first variation of bilinear functionals the resulting algebraic systems of equations are always symmetric. In addition, the transition conditions across the interface are natural. This allows one to approximate the solutions within the interior and exterior regions independently, without imposing any boundary constraints.