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In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. We will derive a formulation as a direct, space-time retarded boundary integral equation, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding parts on the boundary. We prove the continuity and coercivity of the formulation by employing the technique of operational calculus in the Laplace domain. [ABSTRACT FROM AUTHOR]

We consider scalar-valued shape functionals on sets of shapes which are small perturbations of a reference shape. The shapes are described by parameterizations and their closeness is induced by a Hilbert space structure on the parameter domain. We justify a heuristic for finding the best low-dimensional parameter subspace with respect to uniformly approximating a given shape functional. We also propose an adaptive algorithm for achieving a prescribed accuracy when representing the shape functional with a small number of shape parameters. [ABSTRACT FROM AUTHOR]

We construct inverses of the variational electric field boundary integral operator up to compact perturbations on orientable topologically simple screens. We describe them as solution operators of variational problems set in low-regularity standard trace spaces. On flat disks these variational problems do not involve the inversion of any non-local operators. This result lays the foundation for operator preconditioning for the discretized electric field integral equation. [ABSTRACT FROM AUTHOR]

We address shape uncertainty quantification for the two-dimensional Helmholtz transmission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate models and statistics of quantities of interest, we apply an adaptive, anisotropic Smolyak algorithm, which allows to attain high convergence rates that are independent of the number of dimensions activated in the parameter space. We also develop a regularity theory with respect to the spatial variable, with norm bounds that are independent of the parametric dimension. The techniques and theory presented in this paper can be easily generalized to any elliptic problem on a stochastic domain. [ABSTRACT FROM AUTHOR]

We consider isotropic scalar diffusion boundary value problems whose diffusion coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains. We allow so-called material junctions where three or more subdomains may abut. We derive a boundary integral equation of the second kind posed on the skeleton of the subdomain partition that involves, as unknown, only one trace function at each point of each interface. We prove the well-posedness of the corresponding boundary integral equations. We also report numerical tests for Galerkin boundary element discretisations, in which the new approach proves to be highly competitive compared to the well-established first kind direct single-trace boundary integral formulation. In particular, GMRES seems to enjoy fast convergence independent of the mesh resolution for the discrete second kind BIE. [ABSTRACT FROM AUTHOR]

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns. [ABSTRACT FROM AUTHOR]

In (X. Claeys and R. Hiptmair, Integral equations on multi-screens. Integral Equ Oper Theory, 77(2):167-197, ) we developed a framework for the analysis of boundary integral equations for acoustic scattering at so-called multi-screens, which are arbitrary arrangements of thin panels made of impenetrable material. In this article we extend these considerations to boundary integral equations for electromagnetic scattering. We view tangential multi-traces of vector fields from the perspective of quotient spaces and introduce the notion of single-traces and spaces of jumps. We also derive representation formulas and establish key properties of the involved potentials and related boundary operators. Their coercivity will be proved using a splitting of jump fields. Another new aspect emerges in the form of surface differential operators linking various trace spaces. [ABSTRACT FROM AUTHOR]

We consider the nonlinear spatially homogeneous Boltzmann equation and its Fourier spectral discretization in velocity space involving periodic continuation of the density and a truncation of the collision operator. We allow discretization based on arbitrary sets of active Fourier modes with particular emphasis on the family of so-called hyperbolic cross approximations. We also discuss an offset method that takes advantage of the known equilibrium solutions. Extending the analysis in Filbet & Mouhot (2011, Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc., 363, 1947–1980), we establish consistency estimates for the discrete collision operators and stability of the semidiscrete evolution. Under an assumption of Gaussian-like decay of the discrete solution, we give a detailed bound for Hs-Sobolev norms of the error due to Fourier spectral discretization. [ABSTRACT FROM AUTHOR]

Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments. [ABSTRACT FROM AUTHOR]

We study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions. Following [A. Buffa and R. Hiptmair, Numer Math, 100, 1-19 ()] we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of [X. Claeys and R. Hiptmair, Commun Pure and Appl Math, 66, 1163-1201 ()] to situations with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In the present article, we develop a new functional framework for the study of scalar wave scattering by objects, called multi-screens, that are arbitrary arrangements of thin panels of impenetrable materials. From a geometric point of view, multi-screens are a priori non-orientable non-Lipschitz surfaces. We use our new framework to study boundary integral formulations of the scattering by such objects. [ABSTRACT FROM AUTHOR]

We consider the multiple point evaluation problem for an n-dimensional space of functions [ − 1,1[↦ℝ spanned by d-variate basis functions that are the restrictions of simple (say linear) functions to tensor product domains. For arbitrary evaluation points this task is faced in the context of (semi-)Lagrangian schemes using adaptive sparse tensor approximation spaces for boundary value problems in moderately high dimensions. We devise a fast algorithm for performing m ≥ n point evaluations of a function in this space with computational cost O( mlog n). We resort to nested segment tree data structures built in a preprocessing stage with an asymptotic effort of O( nlog n). [ABSTRACT FROM AUTHOR]

The celebrated truncated T-matrix method for wave propagation models belongs to a class of the reduced basis methods (RBMs), with the parameters being incident waves and incident directions. The T-matrix characterizes the scattering properties of the obstacles independent of the incident and receiver direc-tions. In the T-matrix method the reduced set of basis functions for representation of the scattered field is constructed analytically, and hence, unlike other classes of the RBM, the T-matrix RBM avoids com-putationally intensive empirical construction of a reduced set of parameters and the associated basis set. However, establishing a convergence analysis and providing practical a priori estimates for reducing the number of basis functions in the T-matrix method has remained an open problem for several decades. In this work we solve this open problem for time-harmonic acoustic scattering in two and three dimen-sions. We numerically demonstrate the convergence analysis and the a priori parameter estimates for both point-source and plane-wave incident waves. Our approach can be used in conjunction with any numeri-cal method for solving the forward wave propagation problem. [ABSTRACT FROM AUTHOR]

We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in 'subdomains') that directly lends itself to operator preconditioning via Calderón projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise $\widetilde{H}^{-1/2}$-distributions over each associated interface. Through the use of interior Calderón projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δ u + ω u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua's theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations. [ABSTRACT FROM AUTHOR]

Vekua operators map harmonic functions defined on domain in $${\mathbb R^{2}}$$ to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907-1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves. [ABSTRACT FROM AUTHOR]

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator - Δ - ω², ω > 0. They include the so-called ultra weak variational formulation from. This paper is concerned with the a priori convergence analysis of PWDG in the ease of p-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121-136], and plane wave approximation theory. [ABSTRACT FROM AUTHOR]

In this article we analyze a finite element method for solving H(div;Ω)-elliptic interface problems in general three-dimensional Lipschitz domains with smooth material interfaces. The continuous problems are discretized by means of lowest order H(div;Ω)-conforming finite elements of the first family (Raviart-Thomas or Nédélec face elements) on a family of unstructured oriented tetrahedral meshes. These resolve the smooth interface in the sense of sufficient approximation in terms of a parameter δ that quantifies the mismatch between the smooth interface and the finite element mesh. Optimal error estimates in the H(div;Ω)-norms are obtained for the first time. The analysis is based on a so-called δ-strip argument, a new extension theorem for H1(div)-functions across smooth interfaces, a novel non-standard interfaceaware interpolation operator, and a perturbation argument for degrees of freedom in H(div;Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution. [ABSTRACT FROM AUTHOR]

We present a systematic study of how to take into account external excitation in the eddy current model. Emphasis is put on mathematically sound variational formulations and on lumped parameter excitation through prescribed currents and voltages. We distinguish between local excitation at known contacts, known generator current distributions and non-local variants that rely on topological concepts. The latter case entails the violation of Faraday's law at so-called cuts and prevents us from reconstructing a meaningful electric field. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Many boundary integral equation methods used in the simulation of direct electro- magnetic scattering of a time-harmonic wave at a perfectly conducting obstacle break down when applied at frequencies close to a resonant frequency of the obstacle. A remedy is offered by special in- direct boundary element methods based on the so-called combined field integral equation. However, hitherto no theoretical results about the convergence of discretized combined field integral equations have been available. In this paper we propose a new combined field integral equation, convert it into variational form, establish its coercivity in the natural trace spaces for electromagnetic fields, and conclude existence and uniqueness of solutions for any frequency. Moreover, a conforming Galerkin discretization of the variational equations by means of divΓ-conforming boundary elements can be shown to be asymptotically quasi-optimal. This permits us to derive quantitative convergence rates on sufficiently fine, uniformly shape-regular sequences of surface triangulations. [ABSTRACT FROM AUTHOR]

Presents a study on scattering of monochromatic electromagnetic waves at a dielectric object with a rough surface. Steps to establish coercivity of finite elements and boundary elements in electromagnetic scattering; Factors responsible for discretization of electromagnetic waves.

Many boundary integral equations for exterior Dirichlet and Neumann boundary value problems for the Helmholtz equation suffer from a motorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. This article presents combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical h-ace spaces. For the exterior Dirichlet problem the main idea is to use suitable regularizing operators in the framework of an indirect method This permits us to apply the classical convergence theory of conforming Galerkin methods. [ABSTRACT FROM AUTHOR]

This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include $\vec H({\rm div}; \Omega$)- and $\vec (H{\bf curl}; \Omega)$-conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation operators can be approximated. The stability of $\vec H({\rm div}; \Omega)$-conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation, is investigated both theoretically and in numerical experiments. [ABSTRACT FROM AUTHOR]

The focus of this paper is on boundary value problems for Maxwell's equations that feature cylindrical symmetry both of the domain Ω⊂ R3 and the data. Thus, by resorting to cylindrical coordinates, a reduction to two dimensions is possible. However, cylindrical coordinates introduce a potentially malicious singularity at the axis rendering the variational problems degenerate. As a consequence, the analysis of multigrid solvers along the lines of variational multigrid theory confronts severe difficulties. Line relaxation in radial direction and semicoarsening can successfully reign in the degeneracy. In addition, the lack of H1-ellipticity of the double- curl operator entails using special hybrid smoothing procedures. All these techniques combined yield a fast multigrid solver. The theoretical investigation of the method relies on blending generalized Fourier techniques and modern variational multigrid theory. We first determine invariant subspaces of the multigrid iteration operator and analyze the smoothers therein. Under certain assumptions on the material parameters we manage to show uniform convergence of a symmetric V-cycle. [ABSTRACT FROM AUTHOR]

Describes an algorithm for constructing triangulated surfaces. Lipschitz-polyhedron in three-dimensional Euclidean space; Eddy currents; Association of edges of the surface mesh with their degrees of freedom; Discrete surface current; Boundary element methods of electromagnetic fields.

For a Lipschitz-polyhedron Ω C ⊂ R³ we consider eigenvalue problems curl α curl u = Au and gradαdiv u = λu, λ > 0, set in H(curl; Ω) and H(div; Ω). They are discretized by means of the conforming finite elements introduced by Nédélec. The preconditioned inverse iteration in its subspace variant is adapted to these problems. A standard multigrid scheme serves as the preconditioner. The main challenge arises from the large kernels of the operators curl and div. However, thanks to the choice of finite element spaces these kernels have a direct representation through the gradients/rotations of discrete potentials. This makes it possible to use a multigrid iteration in potential space to obtain approximate projections onto the orthogonal complements of the kernels. There is ample evidence that this will lead to an asymptotically optimal method. Numerical experiments confirm the excellent performance of the method even on very fine grids. [ABSTRACT FROM AUTHOR]

Considers the electric field integral equation on the surface of polyhedral domains and its Galerkin discretization by means of divergence-conforming boundary elements. Coercive variational problem; Negative possession of the essential regularity properties of Hodge decompositions.

Derives a symmetric finite element method-boundary element method-coupling for the E-based eddy current model. Properties of potentials and boundary integral operators from a Stratton-Chu-type representation formula; Analysis of the electric field in a nonconducting region; Examination of curl-conforming edge elements and divergence-conforming surface elements.

Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus. [ABSTRACT FROM AUTHOR]

We consider a bounded Lipschitz-polyhedron &om; ⊂R[SUP3] of general topology equipped with a tetrahedral triangulation that induces a mesh Λ[SUBh] of the surface δ&om;. We seek a maximal set of surface edge cycles that are independent in H[SUB1](Λ[SUBh],Z) and bounding with respect to the exterior of &om;. We present an algorithm for constructing suitable 1-cycles in Λ[SUBh] : First, representatives of a basis of the homology group H[SUB1](Λ[SUBh], Z) are constructed, merely using the combinatorial description of the surface mesh Λ[SUBh]. Then, a duality pairing based on linking numbers is used to determine those combinations that are bounding with respect to R[SUP3]\&om;. This is the key to circumventing a triangulation of the exterior region R[SUP3]\&om; in the computations. For shape-regular, quasi-uniform families of meshes, the asymptotic complexity of the algorithm is shown to be O(N[SUP2]), where N is the number of edges of Λ[SUBh]. The scheme provides an essential preprocessing step for all boundary element methods for eddy current simulation, which rely on discrete divergence-free vectorfields and their description through stream functions. [ABSTRACT FROM AUTHOR]

We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced. The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semi-coarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at one-dimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid V-cycle are bounded independently of the number of grid levels involved. In addition, the estimates reveal that convergence is also robust with respect to a singular perturbation in one coordinate direction. Finally, we supply numerical evidence that the algorithm performs satisfactorily in settings more general than those covered by the proof. [ABSTRACT FROM AUTHOR]

In the framework of the macroscopic simulation of electrorheological fluids, we present an extension of the classical Bingham model which goes beyond pure shear flows and thus enables the simulation of settings in more complex geometries. Emphasis is on the numerical solution of the resulting nonsmooth minimization problem. We propose the method of augmented Lagrangians combined with an operator-splitting technique which allows to confine the nonlinearity to local, low-dimensional problems. Numerical results are given that illustrate the electrorheological effects for various shear rates and electric field strengths in case of an electrorheological suspension rotating between two revolving cylinders. [ABSTRACT FROM AUTHOR]

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