Space–time Galerkin least-squares method for the one–dimensional advection–diffusion equation

Abstract

The advection–diffusion equation has a long history as a benchmark for numerical methods. The standard Galerkin finite-element method is accurate but lacks sufficient stability, whereas the least-squares method is stable but not accurate for small values of diffusion coefficient. One possibility to overcome it is to use a stabilized finite-element method such as Galerkin least-squares and this is the standard approach taken in this paper. The space–time GLS has been constructed by using both linear and quadratic B-spline shape functions. If advection dominates over diffusion the numerical solution is difficult, especially if boundary layers are to be resolved. The design of stability parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behaviour of the method with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.