Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Explicit expressions for three-dimensional boundary integrals in linear elasticity

On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lamé equation are included to validate the proposed formulae.     

Sparse second moment analysis for elliptic problems in stochastic domains

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain. This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.     

A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks

The numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered. Since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods, based on global interpolation by analytic functions, show low accuracy. In order to circumvent this issue, a meshfree modification of the method of fundamental solutions is developed, where the approximation basis is enriched by an extra span of corner adapted non-smooth shape functions. The high accuracy of the new method is illustrated by solving several boundary value problems for the Helmholtz equation, modelling physical phenomena from the fields of room acoustics and acoustic resonance.     

On the super-approximation property of Galerkin's method with finite elements

Summary For Galerkin's method with finite elements as trial functions for strongly elliptic operator equations in the Hilbert scaleH ^t the super-approximation property and the optimal convergence rate are obtained by using the Aubin-Nitsche lemma. This applies in particular to spline collocation methods for a wide class of pseudodifferential equations.Dedicated to the memory of Professor Lothar Collatz     

A collocation method for high-frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

Worst case scenario analysis for elliptic problems with uncertainty

This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

Dirac delta methods for Helmholtz transmission problems

In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms.      

Mixed boundary integral methods for Helmholtz transmission problems

In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface.     

The Galerkin scheme for Lavrentiev’sm-times iterated method to solve linear accretive Volterra integral equations of the first kind

In this paper, for the numerical solution of linear accretive Volterra integral equations of the first kind in Hilbert spaces we consider the Galerkin scheme for Lavrentiev’sm-times iterated method, i.e., for each parameter choice for Lavrentiev’sm-times iterated method the arisingm stabilized equations are discretized by the Galerkin scheme. An associated discrepancy principle as parameter choice strategy for this finite-dimensional version of Lavrentiev’sm-times iterated method is proposed, and corresponding convergence results are provided.     

Numerical implementation of the EDEM for modified Helmholtz BVPs on annular domains

In a recent paper by the current authors a new methodology called the Extended-Domain-Eigenfunction-Method (EDEM) was proposed for solving elliptic boundary value problems on annular-like domains. In this paper we present and investigate one possible numerical algorithm to implement the EDEM. This algorithm is used to solve modified Helmholtz BVPs on annular-like domains. Two examples of annular-like domains are studied. The results and performance are compared with those of the well-known boundary element method (BEM). The high accuracy of the EDEM solutions and the superior efficiency of the EDEM over the BEM, make EDEM an excellent alternate candidate to use in the animation industry, where speed is a predominant requirement, and by the scientific community where accuracy is the paramount objective.     

Boundary-variation solution of eigenvalue problems for elliptic operators

We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated far outside their radii of convergence—by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

A composite Level Set and Extended-Domain-Eigenfunction Method for simulating 2D Stokes flow involving a free surface

In this paper, the Extended-Domain-Eigenfunction-Method (EDEM) is combined with the Level Set Method in a composite numerical scheme for simulating a moving boundary problem. The liquid velocity is obtained by formulating the problem in terms of the EDEM methodology and solved using a least square approach. The propagation of the free surface is effected by a narrow band Level Set Method. The two methods both pass information to each other via a bridging process, which allows the position of the interface to be updated. The numerical scheme is applied to a series of problems involving a gas bubble submerged in a viscous liquid moving subject to both an externally generated flow and the influence of surface tension.     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.