Numerical dispersion in the vector finite element method using first-order quadrilateral elements

Although the vector finite element method has been widely used, the errors associated with it have not been thoroughly investigated. For finite elements, one of the most significant errors arises from the inability of the polynomial basis functions to represent the fields exactly within an element. A plane wave propagating through a finite element mesh will experience numerical dispersion as a result of this error. For practical node densities, this numerical dispersion can be characterized by a cumulative phase error. The numerical dispersion of a time-harmonic plane wave propagating through an infinite, two-dimensional, first-order vector finite element mesh composed of uniform quadrilateral elements is investigated. It is shown that the phase error for the vector finite elements is dependent upon the direction of propagation through the mesh and the electrical size of the elements. In addition, a simple formula is given which is an approximation to the exact numerical dispersion.