A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems

Abstract

A variational inequality (VI) and a mixed formulation for an elliptic obstacle problem are considered. Both formulations are discretized by an hp-FE interior penalty discontinuous Galerkin (IPDG) method. In the case of the mixed method, the discrete Lagrange multiplier is a linear combination of biorthogonal basis functions. In particular, also the discrete problems are equivalent. For these formulations a residual based a posteriori error estimate and a hierarchical a posteriori error estimate are derived. For the mixed method the residual based estimate is constructed explicitly, for which the approximation error is split into a discretization error of a linear variational equality problem and additional consistency and obstacle condition terms. For the VI-method a hierarchical estimate based on an overkill solution is derived explicitly. This includes the h–h/2 and p–(p+1)-estimates. The numerical experiments show exponential convergence up to the desired tolerance.