An Element Decomposition Method for the Helmholtz Equation (Communications in Computational Physics, 2020)
It is well-known that the traditional full integral quadrilateral element failsto provide accurate results to the Helmholtz equation with large wave numbers due tothe "pollution error" caused by the numerical dispersion. To overcome this deficiency,this paper proposed an element decomposition method (EDM) for analyzing 2D acousticproblems by using quadrilateral element. In the present EDM, the quadrilateralelement is first subdivided into four sub-triangles, and the local acoustic gradient ineach sub-triangle is obtained using linear interpolation function. The acoustic gradientfield of the whole quadrilateral is then formulated through a weighted averagingoperation, which means only one integration point is adopted to construct the systemmatrix. To cure the numerical instability of one-point integration, a variation gradientitem is complemented by variance of the local gradients. The discretized systemequations are derived using the generalized Galerkin weak form. Numerical examplesdemonstrate that the EDM can achieves better accuracy and higher computational efficiency.Besides, as no mapping or coordinate transformation is involved, restrictionson the shape elements can be easily removed, which makes the EDM works well evenfor severely distorted meshes.