Using rectangular image elements in the SDFEM for a convection–diffusion problem with a boundary layer Original Research Article

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 QpQp, i.e., are tensor products of polynomials of degree p in one variable, where p>1p>1. The error bound ‖INu−uNSD‖⩽CN−(p+1/2)‖INu−uN‖SD⩽CN−(p+1/2) is proved; here uNuN is the computed SDFEM solution, INuINu is chosen in the finite element space to be a special approximant of the true solution u, and ‖⋅SD‖‖⋅‖SD is the streamline-diffusion norm. This result is compared with previously known results for the case p=1p=1. The error bound is a superclose result; uNuN can be enhanced using local postprocessing to yield a modified solution View the MathML sourceu˜N for which View the MathML source‖u−u˜N‖SD⩽CN−(p+1/2).