Explicit error bounds for the α-quasi-periodic Helmholtz problem.

Area of Science:

• Computational electromagnetics
• Applied mathematics
• Wave physics

Background:

• Modeling electromagnetic waves in periodic structures is crucial for optics and photonics.
• Existing methods often assume continuity of the Dirichlet-to-Neumann map, lacking explicit parameter dependencies.
• The finite element method (FEM) offers geometric flexibility but requires rigorous error analysis.

Purpose of the Study:

• To develop and analyze a finite element approach for electromagnetic wave modeling in periodic diffraction gratings.
• To derive a priori error estimates for the α-quasi-periodic transformation.
• To explicitly show the dependence of the error estimate on system parameters like wavenumber.

Main Methods:

• Utilized a variational formulation to examine the well-posedness of the continuous Helmholtz problem.
• Discretized the problem and derived a rigorous a priori error estimate for the α-quasi-periodic solution.
• Obtained an explicit dependence of the regularity constant on the wavenumber and polynomial basis degree.

Main Results:

• Established a rigorous a priori error estimate for the α-quasi-periodic transformation in FEM for diffraction gratings.
• Demonstrated an explicit dependency of the error estimate on the wavenumber and the degree of the polynomial basis.
• Numerical results using the α-quasi-periodic transformation were compared with a lattice sum technique.

Conclusions:

• The derived a priori error estimate guarantees the uniqueness and improves the accuracy of the approximate solution.
• This work addresses limitations in previous studies by providing explicit parameter dependencies in the error analysis.
• The finite element method with the α-quasi-periodic transformation offers a robust approach for analyzing periodic structures.