Parabolic finite element equations in nonconvex polygonal domains

Let Q be a bounded nonconvex polygonal domain in the plane. Consider the initial boundary value problem for the heat equation with homogeneous Dirichlet boundary conditions and semidiscrete and fully discrete approximations of its solution by piece-wise linear finite elements in space. The purpose of this paper is to show that known results for the stationary, elliptic, case may be carried over to the time dependent parabolic case. A special feature in a polygonal domain is the presence of singularities in the solutions generated by the corners even when the forcing term is smooth. These cause a reduction of the convergence rate in the finite element method unless refinements are employed.