Using rectangular Qp elements in the SDFEM for a convection–diffusion problem with a boundary layer

The streamline diffusion finite element method (SDFEM; the method is also known as SUPG) is applied to a convection–diffusion problem posed on the unit square whose solution has exponential boundary layers. A rectangular Shishkin mesh is used. The trial functions in the SDFEM are piecewise polynomials that lie in the space Qp, i.e., are tensor products of polynomials of degree p in one variable, where p>1. The error bound ‖INu−uN‖SD⩽CN−(p+1/2) is proved; here uN is the computed SDFEM solution, INu is chosen in the finite element space to be a special approximant of the true solution u, and ‖⋅‖SD is the streamline-diffusion norm. This result is compared with previously known results for the case p=1. The error bound is a superclose result; uN can be enhanced using local postprocessing to yield a modified solution for which .