Interaction of boundary layers and corner singularities (Q1001789)

The authors consider a constant coefficient elliptic problem in the unit square, physically a convection-diffusion-reaction problem. They are interested in interaction of the boundary layers due to a small diffusion coefficient with the corner singularities. After analysing the behaviour of the exact solution and its relation to a boundary element with support in only one element, they consider, on an equidistant mesh, a \(Q_1\)-finite element space enriched by this boundary element. They prove that, e.g., \(H_1\) approximation is possible when \(h=o(\varepsilon^{1/2})\), instead of \(h=o(\varepsilon^{3/2})\) in other approaches. A series of numerical experiments verifies this conclusion (for \(\varepsilon\) up to \(10^{-15}\)) and illustrates also that the numerical solution detoriates for small \(\varepsilon\) on a fixed mesh. The case of a diffusion-convection problem is not handled fully in the analysis.