Plane wave approximation of homogeneous Helmholtz solutions
Moiola, A., Hiptmair, R. and Perugia, I. (2011) Plane wave approximation of homogeneous Helmholtz solutions. Zeitschrift für angewandte Mathematik und Physik, 62 (5). pp. 809-837. ISSN 0044-2275
To link to this item DOI: 10.1007/s00033-011-0147-y
In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω 2 u = 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.