研究目的
Investigating the finite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML).
研究成果
The paper establishes quasi-optimal a priori error estimates for finite element approximations to the Helmholtz equation in waveguides using PML, demonstrating exponential convergence of PML errors with respect to the width and strength of PML. Numerical experiments confirm the theoretical convergence analysis.
研究不足
The study acknowledges the lack of full regularity of PML solutions and the anisotropic nature of the PML problem, especially with large PML damping parameters. The convergence rate may be affected by the presence of near-cutoff modes.
1：Experimental Design and Method Selection:
The study employs a perfectly matched layer (PML) defined by a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components, truncated with a Neumann condition on an artificial boundary.
2：Sample Selection and Data Sources:
The problem is set in a semi-infinite waveguide with a bounded cross-section, and the wave source is supported in a specific region.
3：List of Experimental Equipment and Materials:
Finite element meshes, specifically anisotropic meshes in the PML regions, are used to handle the anisotropy of the PML problem.
4：Experimental Procedures and Operational Workflow:
The study involves setting up the PML problem, applying finite element approximations, and analyzing the convergence of solutions.
5：Data Analysis Methods:
Quasi-optimal a priori error estimates are established, and numerical experiments are conducted to confirm the convergence analysis.
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