Solving short wave problems using special finite elements - Towards an adaptive approach

Special finite elements capable of containing many wavelengths per nodal spacing are developed and used to solve problems such as the diffraction of a plane wave by a rigid body. The method used, called the Partition of Unity Method, is due to Melenk and Babuska. It consists of incorporating analytical information into a piecewise Galerkin approach. In the case of the Helmholtz problem, the wave potential in two dimensional space is approximated by systems of plane waves propagating in all directions. This leads to larger finite element matrices. However since the elements can contain many wavelengths, the dimension of the global system to be solved is greatly reduced. In a previous work, these special elements were developed by using the same approximating plane waves at all nodes. In this paper, the number and the directions of plane waves can vary from one node to another. It is shown that by taking only a few plane waves clustered around the main directions of propagation, the numerical results converge to the exact solution. This is a step towards developing self adaptive finite elements for short wave problems.