The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains

Abstract We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question ‘how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?’ has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper we prove that for nontrapping problems solved using straight elements the geometric error is of order $kh$, which is then less than the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, we prove that, even for problems with strong trapping, using degree four (in two dimensions) or degree five (in three dimensions) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.