Numerical dispersion in the finite element method

The numerical dispersion in the finite element (FE) method due to the spatial discretisation of the scalar Helmholtz equation is investigated. It is widely understood that a plane wave will propagate along a uniform FE mesh; however, it propagates at the wrong velocity yielding a progressive phase error in the FE solution. This phase dispersion is particularly troublesome for electrically large problems, since it is cumulative and builds up to larger and larger values the farther the wave propagates. In many types of problems, the phase dispersion is the largest source of error. The phase dispersion can be reduced by either increasing the node density of the mesh or by increasing the order of the elements. The magnitude of this error must be known in order to make an informed decision on the appropriate node density and order of element to use. The phase dispersion is studied for first- through fourth-order, one-dimensional elements as a function of the node density. The phase dispersion is shown to decrease rapidly with the increasing order of the elements. The one-dimensional results are indicative of what to expect in the two- and three-dimensional cases.