Some remarks on the behaviour of the finite element solution in nonsmooth domains

In this work, we consider the behaviour of the residual error using a smooth finite element solution for elliptic problems on nonconvex and nonsmooth domains. It is proved that, against expectations, the residual error is unbounded and actually diverges to infinity as the mesh size goes to zero. A numerical example which illustrates this phenomenon will be presented for the Poisson equation on an L-shaped domain using a C¹ Hermite element, and similar results will be shown for a C⁰ element with a posteriori smoothing.