Plane wave approximation of homogeneous Helmholtz solutions
In this article, the approximation of solutions of homogeneous Helmholtz equations \(\Delta u+\omega^2u=0\) with constant coefficients and wave number \(\omega>0\) is considered using linear combinations of plane waves with different directions. The authors combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials that are obtained from Vekua's theory. They focus specifically on the estimates for the approximation of generalized harmonic polynomials by plane waves. In addition, best approximation error estimates in Sobolev norms are derived. Those are explicit in terms of the domain size, the number of plane waves, and the degree of the generalized polynomials.
scientific article; zbMATH DE number 6134359
scientific article