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

We consider the Dirichlet problem of the indefinite Helmholtz equation in 1D, $u''+k^2u=f$ in $(0,1)$, $u(0)=g_0$, $u(1)=g_1$, with a constant wavenumber $k\in(0,\infty)\backslash\pi\mathbb{N}$ and a source term $f\in H^p_0(0,1)$, $p\ge 4$. We propose an approach based on Fourier analysis to derive wavenumber explicit sharp estimates of absolute and relative errors of \emph{finite difference} methods. Such results have been well known for \emph{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 $k(kh)^2/\sigma_k\le4/(\pi-2)$ with $\sigma_k:=\operatorname{dist}(k,\pi\mathbb{N})$, that the worst case attainable convergence order of the absolute error with $\sum_{p=0}^4k^{-p}\|f^{(p)}\|_{L^2}=O(1)$ (or $|g_i|\asymp k^{-1}$) is $(kh)^2/\sigma_k^2$ in the $L^2$-norm and $k(kh)^2/\sigma_k^2$ in the $H^1$-semi-norm, and that of the relative error is $k(kh)^2/\sigma_k$ in both $L^2$- and $H^1$-semi-norms if $\|u^{(p)}\|_{L^2}/\|u^{(p-2)}\|_{L^2}\asymp k^2$ 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.