Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis

Summary. The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data u_D by functions u_D,h in the trace space of a finite element space on _D. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of u_D,h, namely the nodal interpolation and the orthogonal projection in L²(_D) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H¹ and L² a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.Mathematics Subject Classification (1991): 65N30, 65R20, 73C50This work was initiated while C. Carstensen was visiting the Max Planck Institute for Mathematics in the Sciences, Leipzig. S. Bartels acknowledges support by the German Research Foundation (DFG) within the Graduiertenkolleg Effiziente Algorithmen und Mehrskalenmethoden and the priority program Analysis, Modeling, and Simulation of Multiscale Problems. G. Dolzmann gratefully acknowledges partial support by the Max Planck Society and by the NSF through grant DMS0104118.