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 functions and they are calculated from simple algebraic recursion relations. The basis for their selection is that the given (coarse) mesh may duplicate the solutions obtained at common nodes of a finer virtual mesh. If the fine mesh is sufficiently refined, then the coarse mesh solutions converge to the exact solution. The finer mesh is virtual because its associated system of discrete equations is never solved. This multilevel PG method is extended to cases of the non-homogeneous problem with polynomial force functions. The examples considered confirm that this method is successful in accelerating the rate of convergence of the solution even when the force terms are non-polynomial. The multilevel PG method is therefore efficient and powerful for the general non-homogeneous convection-diffusion equation.