Optimal Test Function Petrov-Galerkin Method

Numerical analysis of convection-dominated transport problems are challenging because of dual characteristics of the governing equation. In the finite element method, a strategy is to modify the test function to weight more in the upwind direction. This is called as the Petrov-Galerkin method. In this paper, both N+1 and N+2 Petrov-Galerkin methods are applied to transport problems at high grid Peclet number. Frequency fitting algorithm is used to obtain optimal levels of N+2 upwinding, and the results are discussed. Also, a new Petrov-Galerkin method, named as Optimal Test Function Petrov-Galerkin Method, is proposed in this paper. The test function of this numerical method changes its shape depending upon relative strength of the convection to the diffusion. A numerical experiment is carried out to demonstrate the performance of the proposed method.