Fourier Analysis of Finite Difference Schemes for the Helmholtz Equation in 1D with Dirichlet  Conditions: Sharp Estimates and Relative Errors

We propose an approach based on Fourier analysis to wavenumber explicit sharp estimation of absolute and relative errors of finite difference methods for the Helmholtz equation {in 1D with Dirichlet boundary conditions} and general source terms. We use the approach to analyze the classical centred scheme. For the Fourier interpolants of the discrete solution with homogeneous (or inhomogeneous) Dirichlet conditions, we show rigorously that the worst case attainable convergence order of the absolute error is $k^2h^2$ (or $k^3h^2$) in the $L^2$-norm and $k^3h^2$ (or $k^4h^2$) in the $H^1$-semi-norm, and that of the relative error is $k^3h^2$ in both $L^2$- and $H^1$-semi-norms. Even though the classical centred scheme is well-known, it is the first time that such sharp estimates of absolute and relative errors are obtained. 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.