MATHEMATICS OF COMPUTATION Volume 78, Number 267, July 2009, Pages 1353–1374 S 0025-5718(08)02183-2 Article electronically published on September 25, 2008 NITSCHE’S METHOD FOR GENERAL BOUNDARY CONDITIONS MIKA JUNTUNEN AND ROLF STENBERG Abstract. We introduce a method for treating general boundary conditions in the finite element method generalizing an approach, due to Nitsche (1971), for approximating Dirichlet boundary conditions. We use Poisson’s equations as a model problem and prove a priori and a posteriori error estimates. The method is also compared with the traditional Galerkin method. The theoreti- cal results are verified numerically. 1. Introduction In his classical paper [6] Nitsche discusses techniques for incorporating Dirich- let boundary conditions in the finite element approximation of the model Poisson problem: find u such that −∆u = f in Ω, (1.1) u = u 0 on Γ = ∂Ω. (1.2) Before introducing his technique he discusses the penalty method, i.e. the Ritz ap- proximation to the “perturbed” problem in which the Dirichlet boundary condition (1.2) is replaced by the condition (1.3) ∂u ∂n = 1 ǫ (u 0 − u) on Γ, where ǫ> 0 is a small parameter. He points out the drawbacks of this approach, i.e. nonconformity, which requires a coupling of the penalty parameter to the mesh size, and the possible ill-conditioning of the discrete system when the penalty parameter is too small (see [2] for a recent survey on this). If instead of the Dirichlet problem we consider the problem with the boundary condition (1.3), then the solution to the continuous problem converges to the so- lution of the Dirichlet problem when ǫ → 0. For the finite element discretization the discrete problem gets more ill-conditioned when ǫ approaches zero. In the limit ǫ = 0, we have to switch to some other way of imposing the Dirichlet condition, like the conventional approach or Nitsche’s technique. The following question arises quite naturally: can we extend Nitsche’s method so that it can be used for the whole range of boundary conditions ǫ ≥ 0 ? The purpose of this paper is to give a Received by the editor October 17, 2007, and, in revised form, May 21, 2008. 2000 Mathematics Subject Classification. Primary 65N30. This work was supported by the Finnish National Graduate School in Engineering Mechanics, by the Academy of Finland, and TEKES, the National Technology Agency of Finland. c 2008 American Mathematical Society 1353 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use