Higher order FEM for the obstacle problem of the p-Laplacian

Abstract: 

We consider two higher order finite element discretizations of an obstacle problem with the p-Laplacian differential operator for the singular case 1<p<2 and for the degenerated case 2<p<oo, i.e., for 1<p<oo. The first approach is a standard but non-linear variational inequality formulation in the primal variable u only. The second approach is a primal-dual mixed formulation where the dual variable represents the signed residual of the variational inequality. These two formulations are equivalent and, under mild assumptions on the obstacle, even on the discrete level when using biorthogonal basis functions for the dual variable.

We prove an a priori error estimates as well as a single a posteriori error estimate valid for both formulations. The a posteriori error estimate consists of the same quantities known from the p=2 case. Namely, internal residual and edge residual coming from the variational equation part, and two consistency contributions and violation of the complementarity condition.

Numerical results demonstrating the improved convergence rates of adaptive schemes (mesh size adaptivity with and without polynomial degree adaptation) for p=1.5 and p=3 are presented.