Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers (Q2752293)

The authors use the streamline-diffusion finite element method (SDFEM) with piecewise bilinear trial functions on a Shiskhin mesh to solve the singularly perturbed boundary value problem \(-\varepsilon \triangle u+b\cdot \nabla u+cu=f\) on \(\Omega=(0,1)^2, \;u=0\) on \(\partial\Omega\), where \(\varepsilon\) is a small positive parameter, \( b(x,y)=(b_1(x,y),b_2(x,y))>(\beta_1,\beta_2)>(0,0),\;c(x,y)\geq 0\) for all \((x,y)\in \overline {\Omega}\) and \( c(x,y)- 1/2\) \text{ div} \(b(x,y)\geq c_0>0\) on \(\overline\Omega\), \(\beta_1 \;\text{and} \beta_2\) and \(c_0\) are some constants. The convergence behaviour both in the usual SDFEM norm and in the \(L^{\infty}\) norm is analysed. An \(L^{\infty}\)-norm error estimate is presented. As a corollary the authors prove that the method is convergent in the local \(L^{\infty}\) norm on the fine part of the mesh (i.e. inside the boundary layers) and give the local \(L^{\infty}\)-estimates within the layers.