A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media

被引:65
作者
Gillman, Adrianna [1 ]
Barnett, Alex H. [1 ]
Martinsson, Per-Gunnar [2 ]
机构
[1] Dartmouth Coll, Dept Math, Hanover, NH 03755 USA
[2] Univ Colorado, Dept Appl Math, Boulder, CO 80309 USA
基金
美国国家科学基金会;
关键词
Scattering; Multidomain spectral method; Direct solver; High-order discretization; Second kind Fredholm equation; Boundary integral equation; Nystrom discretization; Impedance-to-Impedance map; FAST DIRECT SOLVER; STRUCTURED LINEAR-SYSTEMS; HELMHOLTZ-EQUATION; WAVE-PROPAGATION; BOUNDARY-CONDITIONS; FINITE-ELEMENT; COMPUTATION;
D O I
10.1007/s10543-014-0499-8
中图分类号
TP31 [计算机软件];
学科分类号
081202 ; 0835 ;
摘要
This paper presents a direct solution technique for the scattering of time-harmonic waves from a bounded region of the plane in which the wavenumber varies smoothly in space. The method constructs the interior Dirichlet-to-Neumann (DtN) map for the bounded region via bottom-up recursive merges of (discretization of) certain boundary operators on a quadtree of boxes. These operators take the form of impedance-to-impedance (ItI) maps. Since ItI maps are unitary, this formulation is inherently numerically stable, and is immune to problems of artificial internal resonances. The ItI maps on the smallest (leaf) boxes are built by spectral collocation on tensor-product grids of Chebyshev nodes. At the top level the DtN map is recovered from the ItI map and coupled to a boundary integral formulation of the free space exterior problem, to give a provably second kind equation. Numerical results indicate that the scheme can solve challenging problems 70 wavelengths on a side to 9-digit accuracy with 4 million unknowns, in under 5 min on a desktop workstation. Each additional solve corresponding to a different incident wave (right-hand side) then requires only 0.04 s.
引用
收藏
页码:141 / 170
页数:30
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