Search

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as k d - 1 / 2 , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as k d to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation. [ABSTRACT FROM AUTHOR]

The method of fundamental solutions has been mainly applied to wave scattering problems in bounded domains and to our knowledge there have not been works addressing density results for general shapes, or addressing the calculation of the complex resonance frequencies that occur in exterior problems. We prove density and convergence of the fundamental solutions approximation in the context of wave scattering problems, with and without a priori knowledge of the frequency, which is of particular importance to detect resonance frequencies for trapping domains. We also present several numerical results that illustrate the good performance of the method in the calculation of complex resonance frequencies for trapping and non trapping domains in 2D and 3D. [ABSTRACT FROM AUTHOR]

We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is well-defined in a tensor product of appropriate local function spaces, each with L 2 impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software. [ABSTRACT FROM AUTHOR]

For the h-finite-element method (h-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth h must decrease with the frequency k to maintain accuracy as k increases has been studied since the mid 80's. Nevertheless, there still do not exist in the literature any k-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, p, equal to one), the condition " h 2 k 3 sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of k) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for p ≥ 2 . A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures. [ABSTRACT FROM AUTHOR]

We construct a new Absorbing Boundary Condition (ABC) adapted to solving the Helmholtz equation in polygonal domains in dimension two. Quasi-continuity relations are obtained at the corners of the polygonal boundary. This ABC is then used in the context of domain decomposition where various stable algorithms are constructed and analysed. Next, the operator of this ABC is adapted to obtain a transmission operator for the Domain Decomposition Method (DDM) that is well suited for broken line interfaces. For each algorithm, we show the decrease of an adapted quadratic pseudo-energy written on the skeleton of the mesh decomposition, which establishes the stability of these methods. Implementation within a finite element solver (GMSH/GetDP) and numerical tests illustrate the theory. [ABSTRACT FROM AUTHOR]

We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or the polynomial degree, though the obtained guaranteed bound may lead to large error overestimation. We next demonstrate that the estimator is locally efficient, robust in all regimes with respect to the polynomial degree, and asymptotically robust with respect to the wavenumber. Finally we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp. [ABSTRACT FROM AUTHOR]

Helmholtz wave scattering by open screens in 2D can be formulated as first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners in the form of square-roots of on-curve differential operators both for the Dirichlet and Neumann boundary conditions on the screen. They generalize the so-called "analytical" preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples. [ABSTRACT FROM AUTHOR]

This paper aims at reviewing and analysing the method of reflections, which is an iterative procedure designed for solving linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on boundary integral representation. For both the sequential and parallel forms of the method appearing in the literature, we interpret the procedure in terms of projection operators. Using a Hilbert space setting and orthogonality, we prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. Several examples of boundary value problems that enter such a framework are given, an alternative proof of convergence is provided in a case which does not. A few numerical tests conclude the study. [ABSTRACT FROM AUTHOR]

The Poisson–Nernst–Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserves the steady state). Solvability of the semi-discrete scheme is proved and a simple fixed point iteration is proposed to solve the fully discrete scheme. Numerical examples in both 1D and 2D and for multiple species are presented to demonstrate the convergence and properties of the proposed scheme. [ABSTRACT FROM AUTHOR]

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the h-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k → ∞ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how h must decrease with k to maintain k-independent quasi-optimality of the Galerkin solutions as k → ∞ when the obstacle is nontrapping. [ABSTRACT FROM AUTHOR]

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation can be solved by replacing the integral operator with a finite difference calculation on a regular grid. A practical method of this type has been developed by the second author. In this paper we prove the validity of a simplified version of this method for the Dirichlet problem in a general domain in R2 or R3. Given a boundary value, we solve for a discrete version of the density of the double layer potential using a low order interface method. It produces the Shortley-Weller solution for the unknown harmonic function with accuracy O(h2). We prove the unique solvability for the density, with bounds in norms based on the energy or Dirichlet norm, using techniques which mimic those of exact potentials. The analysis reveals that this crude method maintains much of the mathematical structure of the classical integral equation. Examples are included. [ABSTRACT FROM AUTHOR]

We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and post-multiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity. [ABSTRACT FROM AUTHOR]

There has been much recent research on preconditioning discretisations of the Helmholtz operator $$\Delta + k^2 $$ (subject to suitable boundary conditions) using a discrete version of the so-called 'shifted Laplacian' $$\Delta + (k^2+ \mathrm{i}\varepsilon )$$ for some $$\varepsilon >0$$ . This is motivated by the fact that, as $$\varepsilon $$ increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large $$\varepsilon $$ can be so that the shifted problem provides a preconditioner that leads to $$k$$ -independent convergence of GMRES, and our main result is a sufficient condition on $$\varepsilon $$ for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the sound-soft scattering problem (with the radiation condition in the latter problem imposed as a far-field impedance boundary condition). Note that we do not address the important question of how large $$\varepsilon $$ should be so that the preconditioner can easily be inverted by standard iterative methods. [ABSTRACT FROM AUTHOR]

In this article, the construction of nested bases approximations to discretizations of integral operators with oscillatory kernels is presented. The new method has log-linear complexity and generalizes the adaptive cross approximation method to high-frequency problems. It allows for a continuous and numerically stable transition from low to high frequencies. [ABSTRACT FROM AUTHOR]

We present a nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many finite volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero. Finally we present numerical results showing that these corrections suppress local minima produced by the original finite volume scheme. [ABSTRACT FROM AUTHOR]

We are interested in the problem of a bidimensional acoustic wave propagation in a medium including a small obstacle with homogeneous Dirichlet boundary condition. We present and analyse a numerical scheme suitable for finite elements that does not suffer from numerical locking, and takes the presence of the small obstacle into account. It is based on the fictitious domain method combined with matched asymptotic expansions. [ABSTRACT FROM AUTHOR]

Summary. A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre-Galerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. [ABSTRACT FROM AUTHOR]

In this paper a study of the coupling between integral equations and finite element methods is presented for two problems of propagation in frequency domain. It is shown that these problems can be viewed as multidomain problems and treated by the mean of the Schur complement technique. The complement coming from the integral equation part is expressed with the integral operators of the scattering theory. This allows to predict the behaviour of the Schur method, either primal or dual, as far as its convergence speed is concerned. Furthermore, the difference of behaviour between electromagnetism and acoustics from this point of view is explained. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergence O ( h) is obtained. [ABSTRACT FROM AUTHOR]

We consider the two-dimensional Helmholtz equation Δ u+λ u=0 in D with the boundary conditions u=0 on ∂ D. D is the 'Swiss Cross' - a region consisting of five unit squares. A method based on the concept of 'Coherence' is utilized to determine an approximation for the first eigenvalue λ= λ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for λ, and to test some conjectures on its relations with torsional rigidity. [ABSTRACT FROM AUTHOR]