On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part II – Analysis for P1P1 and Q1Q1 finite elements

An unwelcome feature of the popular streamline upwind/Petrov–Galerkin (SUPG) stabilization of convection-dominated convection–diffusion equations is the presence of spurious oscillations at layers. A review and a comparison of the most methods which have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers are given in Part I, [V. John, P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review, Comput. Methods Appl. Mech. Engrg. 196 (2007) 2197–2215]. In the present paper, the most promising of these SOLD methods are investigated in more detail for P1P1 and Q1Q1 finite elements. In particular, the dependence of the results on the mesh, the data of the problems and parameters of the methods are studied analytically and numerically. Furthermore, the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined. Our conclusion is that, also for simple problems, any of the SOLD methods generally provides solutions with non-negligible spurious oscillations.