954 research outputs found

    Spectral method for matching exterior and interior elliptic problems

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    A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green's function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

    Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations

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    We consider three problems for the Helmholtz equation in interior and exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee comments and added several reference

    Boundary integral methods in high frequency scattering

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    In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources

    Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions

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    This paper is concerned with fast and accurate computation of exterior wave equations truncated via exact circular or spherical nonreflecting boundary conditions (NRBCs, which are known to be nonlocal in both time and space). We first derive analytic expressions for the underlying convolution kernels, which allow for a rapid and accurate evaluation of the convolution with O(Nt)O(N_t) operations over NtN_t successive time steps. To handle the onlocality in space, we introduce the notion of boundary perturbation, which enables us to handle general bounded scatters by solving a sequence of wave equations in a regular domain. We propose an efficient spectral-Galerkin solver with Newmark's time integration for the truncated wave equation in the regular domain. We also provide ample numerical results to show high-order accuracy of NRBCs and efficiency of the proposed scheme.Comment: 22 pages with 9 figure

    Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

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    This paper analyses the following question: let Aj\mathbf{A}_j, j=1,2,j=1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations (Ajuj)+k2njuj=f\nabla\cdot (A_j \nabla u_j) + k^2 n_j u_j= -f. How small must A1A2Lq\|A_1 -A_2\|_{L^q} and n1n2Lq\|{n_1} - {n_2}\|_{L^q} be (in terms of kk-dependence) for GMRES applied to either (A1)1A2(\mathbf{A}_1)^{-1}\mathbf{A}_2 or A2(A1)1\mathbf{A}_2(\mathbf{A}_1)^{-1} to converge in a kk-independent number of iterations for arbitrarily large kk? (In other words, for A1\mathbf{A}_1 to be a good left- or right-preconditioner for A2\mathbf{A}_2?). We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients AA and nn. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different AA and nn, and the answer to the question above dictates to what extent a previously-calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices

    A fast and well-conditioned spectral method for singular integral equations

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    We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in O(m2n){\cal O}(m^2n) operations using an adaptive QR factorization, where mm is the bandwidth and nn is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to O(mn){\cal O}(m n) operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface
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