Advances in Computational Mathematics 15: 3–23, 2001.  2002 Kluwer Academic Publishers. Printed in the Netherlands. A posteriori error estimators and adaptivity for finite element approximation of the non-homogeneous Dirichlet problem Mark Ainsworth a,∗ and Donald W. Kelly b a Mathematics Department, Strathclyde University, 26 Richmond Street, Glasgow G1 1XH, Scotland E-mail: M.Ainsworth@strath.ac.uk b School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney 2052, Australia Received 5 February 2001; revised 3 May 2001 Communicated by Y. Xu Techniques are developed for a posteriori error analysis of the non-homogeneous Dirich- let problem for the Laplacian giving computable error bounds for the error measured in the energy norm. The techniques are based on the equilibrated residual method that has proved to be reliable and accurate for the treatment of problems with homogeneous Dirichlet data. It is shown how the equilibrated residual method must be modified to include the practically important case of non-homogeneous Dirichlet data. Explicit and implicit a posteriori error es- timators are derived and shown to be efficient and reliable. Numerical examples are provided illustrating the theory. Keywords: finite element analysis, non-homogeneous Dirichlet problem, a posteriori error estimation, adaptive refinement algorithm AMS subject classification: 65N50, 65N15, 65N30 1. Introduction A posteriori error estimators are essential for reliable and adaptive finite element analysis [3,7]. While many techniques are available for problems with homogeneous essential boundary conditions, most a posteriori error estimators for problems with non- trivial essential boundary conditions either neglect the error due to the use of approx- imate boundary data, or avoid the issue altogether by assuming the Dirichlet data are piecewise linear. However, it is easy to construct examples where the error incurred by approximating the boundary data may constitute the major part of the total error. A reli- able a posteriori error estimator must therefore take account of the error incurred by the use of approximate Dirichlet data. ∗ The work of this author was supported in part by the Engineering and Physical Sciences Research Council of Great Britain under grant GR/L90507.