Coercive combined field integral equations

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 trace 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 method

Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods

We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.Comment: Submitted as a conference proceeding; minor revisio

Non-Reflecting Boundary Conditions for Maxwell's Equations

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equatio

Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderón projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new metho

Multiple traces boundary integral formulation for Helmholtz transmission problems

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 preconditionin

Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems

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 H 1(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 solutio

Multiple point evaluation on combined tensor product supports

We consider the multiple point evaluation problem for an n-dimensional space of functions [ − 1,1[ d ↦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 d n). We resort to nested segment tree data structures built in a preprocessing stage with an asymptotic effort of O(nlog d − 1 n

Vekua theory for the Helmholtz operator

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 wave

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215-235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space metho