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

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    This study introduces a finite element method for modeling electromagnetic waves in diffraction gratings. It derives an error estimate for the α-quasi-periodic transformation, improving accuracy for wave propagation problems.

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    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.