SOLUTION OF THE ONE-DIMENSIONAL CONVECTION-DIFFUSION EQUATION BY A MULTILEVAL PETROV-GALERKIN METHOD

A multilevel Petrov-Galerkin (PG) finite element method to accurately solve the one-dimensional convection-diffusion. equation is presented. In this method, the weight functions. are different from the basis fu nctions and they are calculated from simple algebraic recursion relati ons. The basis for their selection is that the given (coarse) mesh may duplicate the solutions obtained at common nodes of a finer virtual m esh. If the fine mesh is sufficiently refined, then the coarse mesh so lutions converge to the exact solution. The finer mesh is virtual beca use its associated system of discrete equations is never solved. This multilevel PG method is extended to cases of the non-homogeneous probl em with polynomial force functions. The examples considered confirm th at this method is successful in accelerating the rate of convergence o f the solution even when the force terms are non-polynomial. The multi level PG method is therefore efficient and powerful for the general no n-homogeneous convection-diffusion equation.