Plane wave approximation of homogeneous Helmholtz solutions

A. Moiola, R. Hiptmair, I. Perugia

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Abstract

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω2u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua's theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.

Original languageEnglish
Pages (from-to)809-837
Number of pages29
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume62
Issue number5
DOIs
Publication statusPublished - Oct 2011

Austrian Fields of Science 2012

  • 101014 Numerical mathematics

Keywords

  • Approximation by plane waves
  • Generalized harmonic polynomials
  • Homogeneous Helmholtz solutions
  • Jacobi-Anger formula
  • Vekua's theory

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