Petrov—Galerkin method for multidimensional, time‐dependent, convective‐diffusion equations

Abstract

Abstract A Petrov—Galerkin finite element method for two and three‐dimensional time dependent convection‐diffusion equations is presented in the context of two space dimensions. The method involves perturbed weighting functions in the weighted residuals formulation, that are bilinear in space and quadratic in time and depend on two parameters which are calculated according to a local analysis for the one dimensional case. The perturbations to the weighting functions can be interpreted as an added local anisotropic balancing diffusion and an added dispersion in the direction of the convective motion. The effectiveness of the method is shown through several examples involving the convective and diffusive transport of scalar distributions in known velocity fields.