Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L2L2 or H1H1-norm are derived under the assumption that hh, h2ω2h2ω2 and h2ω3h2ω3 are sufficiently small, where hh is the mesh size and ωω denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.