Notes on Petrov-Galerkin Weighting Functions in Connection with the Scalar Convention-Diffusion Equation

摘要

The report is concerned with the finite element method for convection-diffusion boundary value problems in one and two dimensions. The primary goal is to obtain a method giving accurate nodal values. The other purpose is didactic: The authors try to proceed in a way that very little prior knowledge about the subject is needed to understand the lines of thought. The one- and two-dimensional cases are treated separately. In both cases the appropriate weak form is derived starting from the boundary value problem in its strong form. In contrary to the usual treatment the outcome of the manipulations is not only the weak form but in addition detailed information about the appropriate weighting functions. The general one-dimensional case is treated first and a method which could be labelled as a conforming one is obtained. The two-dimensional case turns out to be much more difficult; at least in practice. However, the authors suggest some possible approximate procedures as starting points for further studies. Finally some numerical results are given in the one- and two-dimensional cases. The results are compared with the ones obtained by the Galerkin-, GGLS- and SUPG-methods.