A Fourier analysis and dynamic optimization of the Petrov–Galerkin finite element method

A Fourier analysis of the linear and quadratic N + 1 and N + 2 Petrov–Galerkin finite element methods applied to the one‐dimensional transient convective‐diffusion equation is performed. The results show that a priori optimization of the N + 1 method is not possible because dissipative errors are introduced as dispersive errors are reduced (any optimization is subjective). However, a priori optimization of the N + 2 Petrov–Galerkin method is possible because the reduction of dispersion errors can be accomplished without the addition of artificial dissipation. The Spectrally Weighted Average Phase Error Method (SWAPEM) for the optimization of the N + 2 Petrov–Galerkin method is introduced, in which the N + 2 weighting parameter is chosen at each time step to minimize the integral over wave number of the phase error of Fourier modes, weighted by the frequency content of the global solution at the previous time step (obtained via FFT). The method is dynamic, and general in that the dependence of the weighting parameter on the solution waveform is accounted for. Optimal values predicted by the method are in excellent agreement with those suggested by the numerical experimentation of others. Simulations of the pure convective transport of a Gaussian plume and a triangle wave are discussed to illustrate the effectiveness of the method.