On the order of convergence of a finite element method in regions with curved boundaries

Abstract

Finite element methods using elements with curved sides are frequently applied to obtain approximate solutions of elliptic boundary value problems in a bounded open plane region R (see, for example, [1]--[7]). The finite element method suggested by ZLAMAJ~ (see [7]) has the iml?otant advantage that it gives directly the first partial derivatives of the solution of the boundary value problem. In Zl~imal's paper there are given error bounds for this method. In deriving the error bounds, however, it is assumed that the solution of the boundary value problem has square integrable third or fourth partial derivatives in R. These assumptions are satisfied, in general, only if the boundary of R is sufficiently smooth (see [8]--[9]). In the present paper we shall obtain error bounds for Zl~mal's method without these assumptions.