Inhomogeneous Dirichlet boundary condition in the a posteriori error control of the obstacle problem

We study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous Dirichlet boundary condition. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive a posteriori error bounds which are in simpler form and efficient. To accomplish this, we construct and use a post-processed solution ũh of the discrete solution uh which satisfies the exact boundary conditions sharply although the discrete solution uh may not satisfy. We propose two post processing methods and analyze them, namely the harmonic extension and a linear extension. The theoretical results are illustrated by the numerical results.