Problems in Polygonal Domains

Abstract

In earlier parts of this book we have generally assumed the spatial domain Ω to have a smooth boundary ∂Ω, which has made it possible to guarantee that the solution of the initial-boundary value problem is sufficiently regular for the purpose at hand, provided the data of the problem are sufficiently smooth and satisfy certain compatibility conditions at t = 0. In this chapter we shall consider the case when Ω is a plane polygonal domain. In this case singularities will in general appear in the solution even for smooth compatible data, and this will affect the convergence properties of the approximating finite element solution. We shall analyze in some detail the case of piecewise linear finite elements. In this case, no special difficulties arise whenΩ is convex, but when Ω is nonconvex the singularities will normally reduce the rate of convergence both for elliptic and for parabolic problems.