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 language | English |
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Pages (from-to) | 809-837 |
Number of pages | 29 |
Journal | Zeitschrift fur Angewandte Mathematik und Physik |
Volume | 62 |
Issue number | 5 |
DOIs | |
Publication status | Published - 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