Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides

Abstract

In this paper, we study finite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML). The PML is defined in terms of a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components respectively, and truncated with a Neumann condition on an artificial boundary rather than a Dirichlet condition for cutoff modes that waveguides may allow. In the finite element analysis for the PML problem, we have to deal with two difficulties arising from the lack of full regularity of PML solutions and the anisotropic nature of the PML problem with, in particular, large PML damping parameters. Anisotropic finite element meshes in the PML regions depending on the damping parameters are used to handle anisotropy of the PML problem. As a main goal, we establish quasi-optimal a priori error estimates, that does not depend on anisotropy of the PML problem (when no cutoff mode is involved), including the exponentially convergent PML error with respect to the width and the strength of PML. The numerical experiments that confirm the convergence analysis will be presented.