Errors in Projection of Plane Waves Using Various Basis Functions

Basis functions play important roles in computational electromagnetics (CEM). It is interesting to investigate the errors in the projection of the equivalent current of a plane wave using various basis functions. In this work, the projection error of various basis functions is studied. The basis functions involved are the pulse basis function, the triangular basis function, higher-order versions of these basis functions, and the divergence-conforming basis function on rectangular and triangular elements. The projection errors are derived in closed form. The asymptotic expression of the closed form is given. The analytical results are verified by numerical results. The projection error of the pth order one-dimensional (1D) basis is asymptotically inversely proportional to the (p+1)th power of the density of unknowns. Based on the closed-form projection errors in the one-dimensional case, it is found that when the expansion basis is fixed, the application of different testing functions only affects the coefficient of the projection error, rather than the order. Generally, the error of the divergence-conforming basis in the projection of curl-free vectors is less than that of divergence-free vectors.