The standard finite element method, based on piecewise polynomial Galerkin approximation, is optimal for the Laplace operator in the sense that it minimizes the error in the energy norm (which is the H{sup}1 semi-norm in this case). This property assures good performance of the computation at any mesh refinement, i.e., high coarse-mesh accuracy. Good numerical performance at any mesh refinement is no longer guaranteed for other cases. The Helmholtz operator, describing time-harmonic acoustic and electromagnetic waves, may lose ellipticity with increasing wave number (since the bilinear form no longer induces a norm). This is related to the pollution effect, in which finite element solutions of the Helmholtz equation differ significantly from the best approximation due to spurious dispersion in the computation. In practical terms this leads to an increase in the cost of finite element analysis of the Helmholtz equation at higher wave numbers.
展开▼
