A Bernstein–Bézier finite element method with perfectly matched layer for refraction and diffraction of waves in coastal regions

Abstract

The aim of this paper is to effectively solve wave problems, governed by the linear elliptic mild-slope equation, on unstructured triangular meshes based on the Bernstein–Bézier finite element method. The present model takes into account non-uniform bathymetry and enables to accurately describe wave agitation problems and it copes with the pollution effect. A domain truncation method relying on the Perfectly Matched Layer (PML) concept is performed to address the issues related to open region domains. The proposed PML model uses a non-standard weak form of the truncated mild-slope equation to handle the incident wave field weakly and takes into account external bathymetry effects. A low-complexity procedure, exploiting the tensorial property of Bernstein polynomials in conjunction with the sum factorization method, is applied to set up the local high-order finite element matrices. Additionally, static condensation is applied element-wise to reduce the memory requirements. To avoid further sources of errors due to the interpolation of geometry, an accurate description of curved elements is adopted based on a blending map method. An analysis of $h$-convergence using radial PML is conducted by investigating a wave scattering problem by a circular island, where the bathymetry is initially assumed to be constant, and then including a parabolic shoal. The conditioning of the system matrix is also analyzed. The results clearly demonstrate that Bernstein–Bézier finite element method with radial PML considerably reduces memory requirements while maintaining targeted accuracy. A comparison study in the case of constant bathymetry shows that both radial and Cartesian PMLs yield similar performance in terms of accuracy. To further assess the efficiency of our model, a benchmark dealing with wave scattering by an elliptical shoal is investigated to demonstrate the performance of Cartesian PML with exterior bathymetry effects. Our numerical results are therefore compared with available experimental data as well as those found in the literature.