The enriched quadrilateral overlapping finite elements for time-harmonic acoustics

The pronounced numerical dispersions and numerical anisotropy make solutions from the finite element model using low-order elements unreliable for the time-harmonic acoustic problems with fairly large wavenumber. In this work we propose a novel enriched quadrilateral overlapping elements for Helmholtz problems. In this scheme, the original overlapping elements are strengthened by the harmonic trigonometric functions stemming from the spectral techniques. Since all additional degrees of freedom are aligned on the vertex node of every overlapping element, the proposed method can be directly applied to the original finite element model without changing the mesh topology. Because of the enriched approximation space, the proposed method can significantly suppress the numerical dispersions with practically negligible numerical anisotropy, and can be more computationally efficient in providing comparable solution accuracy compared to the original scheme and the classic finite element method. Besides, the linear dependence issue is completely avoided and these enriched overlapping elements are distortion-insensitive. In this work, the original variational formulation is perturbed using the penalty method to impose the essential boundary conditions. Numerical experiments show that the developed method can reduce user interventions in mesh creation and adjustment, and is promising in practical engineering applications for time-harmonic acoustics.