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An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons. (English) Zbl 1216.65151

A nonpolynomial finite element method for time-harmonic scattering from a sound-soft polygonally shaped bounded obstacle in \(\mathbb R^2\) is analyzed. Exponential convergence bounds with a rate that can be numerically computed using techniques from conformal maps are given [cf. T. Betcke, IMA J. Numer. Anal. 27, No. 3, 451–478 (2007; Zbl 1221.65297)]. The asymptotic convergence rate is wavenumber-independent. It is observed that the overall computation time for this method is competitive with the corresponding method for less to medium frequencies [cf. S. N. Chandler-Wilde and S. Langdon, SIAM J. Numer. Anal. 45, No. 2, 610–640 (2007; Zbl 1162.35020)]. Numerical stability of the method is also discussed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A45 Diffraction, scattering
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Software:

LAPACK; Matlab
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