82 research outputs found
Hierarchical interpolative factorization for elliptic operators: integral equations
This paper introduces the hierarchical interpolative factorization for
integral equations (HIF-IE) associated with elliptic problems in two and three
dimensions. This factorization takes the form of an approximate generalized LU
decomposition that permits the efficient application of the discretized
operator and its inverse. HIF-IE is based on the recursive skeletonization
algorithm but incorporates a novel combination of two key features: (1) a
matrix factorization framework for sparsifying structured dense matrices and
(2) a recursive dimensional reduction strategy to decrease the cost. Thus,
higher-dimensional problems are effectively mapped to one dimension, and we
conjecture that constructing, applying, and inverting the factorization all
have linear or quasilinear complexity. Numerical experiments support this claim
and further demonstrate the performance of our algorithm as a generalized fast
multipole method, direct solver, and preconditioner. HIF-IE is compatible with
geometric adaptivity and can handle both boundary and volume problems. MATLAB
codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat
A Galerkin boundary element method for high frequency scattering by convex polygons
In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains
A new integral representation for quasiperiodic fields and its application to two-dimensional band structure calculations
In this paper, we consider band-structure calculations governed by the
Helmholtz or Maxwell equations in piecewise homogeneous periodic materials.
Methods based on boundary integral equations are natural in this context, since
they discretize the interface alone and can achieve high order accuracy in
complicated geometries. In order to handle the quasi-periodic conditions which
are imposed on the unit cell, the free-space Green's function is typically
replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic
Green's function diverges for families of parameter values that correspond to
resonances of the empty unit cell. Here, we bypass this problem by means of a
new integral representation that relies on the free-space Green's function
alone, adding auxiliary layer potentials on the boundary of the unit cell
itself. An important aspect of our method is that by carefully including a few
neighboring images, the densities may be kept smooth and convergence rapid.
This framework results in an integral equation of the second kind, avoids
spurious resonances, and achieves spectral accuracy. Because of our image
structure, inclusions which intersect the unit cell walls may be handled easily
and automatically. Our approach is compatible with fast-multipole acceleration,
generalizes easily to three dimensions, and avoids the complication of
divergent lattice sums.Comment: 25 pages, 6 figures, submitted to J. Comput. Phy
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Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University
This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield
On-surface radiation condition for multiple scattering of waves
The formulation of the on-surface radiation condition (OSRC) is extended to
handle wave scattering problems in the presence of multiple obstacles. The new
multiple-OSRC simultaneously accounts for the outgoing behavior of the wave
fields, as well as, the multiple wave reflections between the obstacles. Like
boundary integral equations (BIE), this method leads to a reduction in
dimensionality (from volume to surface) of the discretization region. However,
as opposed to BIE, the proposed technique leads to boundary integral equations
with smooth kernels. Hence, these Fredholm integral equations can be handled
accurately and robustly with standard numerical approaches without the need to
remove singularities. Moreover, under weak scattering conditions, this approach
renders a convergent iterative method which bypasses the need to solve single
scattering problems at each iteration.
Inherited from the original OSRC, the proposed multiple-OSRC is generally a
crude approximate method. If accuracy is not satisfactory, this approach may
serve as a good initial guess or as an inexpensive pre-conditioner for Krylov
iterative solutions of BIE
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