Fourier Analysis of Finite Difference Methods for the Helmholtz Equation (Hui Zhang, Xi'an Jiaotong-Liverpool University)

17.06.2025 14:00 – 15:00

Inspired by the work of Dwarka and Vuik on the pollution error, we show that Fourier
analysis can be used as a quantitative tool for estimating the accuracy of finite difference methods
for the Helmholtz equation. In particular, for the classical 3-point central scheme for the 1D
Dirichlet problem, we show rigorously the exact order of convergence in $H^1$-norm is $k^3
h^2$. Previously, Babuska, Sauter et al have established similar results for the linear FEM. We show
sharpness of the order with two-sided bounds, and find also the order of relative errors for nonzero
source problems. As a visual tool, Fourier analysis make it straightforward to draw conclusions from
the curve of error against frequency. For example, we use it to compare the accuracy of some
optimized finite difference methods.

Lieu

Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

entrée libre