Shape calculus and finite element method in smooth domains

The use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary. Hence the approximation error consists of two parts: the geometric part and the finite element part. We propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain. The finite element part of the error can be treated in the standard way. To estimate the geometric part of the error, we need quantitative estimates related to perturbation of the geometry. We derive such estimates using the techniques developed for shape sensitivity analysis. In this paper we consider the dilemma of “smooth polygonal domains” related to error analysis of the finite element method. The dilemma is that the finite element methods are naturally formulated in polygonal (or more generally in piecewise polynomial) geometries. On the other hand, the abstract error estimates rely on interpolation error estimates that require smoothness of the solution that is typically achieved only in regular geometries. In the literature this question has been treated in several ways. Perhaps the most popular approach is that of the above mentioned smooth polygonal domains. That is, the analysis is carried out assuming that the domain is polygonal and hence the grid fits exactly to the domain, and at the same time the solution is regular enough for the optimal interpolation estimates. At the other extreme are the works where the curved boundary is captured more or less exactly by introducing corresponding curved elements, [4], [5], [7], [16]. In this case the analysis can be made rigorous but the price to pay is more complicated local analysis and implementation, especially if the basis functions are modified in the curved elements. If one does not want to modify the basis functions near the boundary, then the approach of [3] provides a way to analyse the resulting variational crime on the boundary. In [7] this approach has been used to construct optimal interpolated essential boundary conditions for curved boundaries. Finally, there exists quite a number of papers where the smooth domain is approximated by a polygonal one which is then triangulated. In many of the papers the error analysis is based on some specific feature, like trivial extension of the finite element solution outside of the polygonal domain. Also the analysis of the Received by the editor November 17, 1997. 2000 Mathematics Subject Classification. Primary 65N30; Secondary 49Q12.