2025/01/28 by Martin J. Gander, Hui Zhang, Gander, Martin J. +2
Mathematics · #65N06 #65N12 #65N15 #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2501.16696
openalex publication_date 2025/01/28 · openalex created_date 2025/01/30 · openalex updated_date 2026/07/28
We consider the Dirichlet problem of the indefinite Helmholtz equation in 1D, u''+k2u=f in (0,1), u(0)=g0, u(1)=g1, with a constant wavenumber k∈(0,∞)\backslashπℕ and a source term f∈ Hp0(0,1), p≥ 4. We propose an approach based on Fourier analysis to derive wavenumber explicit sharp estimates of absolute and relative errors of finite difference methods. Such results have been well known for finite element methods (FEM). We use the approach to analyze the classical centered finite difference scheme. For the Fourier interpolants of the discrete solution with homogeneous (or inhomogeneous) Dirichlet conditions, we show rigorously, under the two assumptions k>20 and k(kh)2/σk≤4/(π-2) with σk:=dist(k,πℕ), that the worst case attainable convergence order of the absolute error with ∑p=04k-p‖f(p)‖L2=O(1) (or |gi|\asymp k-1) is (kh)2/σk2 in the L2-norm and k(kh)2/σk2 in the H1-semi-norm, and that of the relative error is k(kh)2/σk in both L2- and H1-semi-norms if ‖u(p)‖L2/‖u(p-2)‖L2\asymp k2 for p=2,3. In particular, the lower bounds of these error estimates are established rigorously in the same orders as the upper bounds, which is the main novelty of this work. We show also that the Fourier analysis approach can be used as a convenient visual tool for evaluating finite difference schemes in presence of source terms, which is beyond the scope of dispersion analysis. The results from the theory and visual analysis are corroborated by numerical experiments.