Franz, Sebastian; Linß, Torsten; Roos, Hans-Görg; Schiller, Sebastian Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems. (English) Zbl 1224.65246 J. Comput. Math. 28, No. 1, 32-44 (2010). Summary: The edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square by using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for \(\|\pi u-u^h\|_E\), where \(\pi u\) is some interpolant of the solution \(u\) and \(u^h\) is the discrete solution. This supercloseness result implies an optimal error estimate with respect to the \(L_2\)-norm and opens the door to the application of postprocessing for improving the discrete solution. Cited in 1 ReviewCited in 8 Documents MSC: 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations Keywords:convection-diffusion problems; edge stabilization; uniform convergence; Shishkin mesh; finite element method; superconvergence; exponential boundary layers; error bound PDFBibTeX XMLCite \textit{S. Franz} et al., J. Comput. Math. 28, No. 1, 32--44 (2010; Zbl 1224.65246) Full Text: DOI Link