2018/11/22 by Hafemeyer, Dominik, Kahle, Christian, Pfefferer, Johannes
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1811.09260
This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the L2-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem. The underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of point-wise error estimates results in a rate less than two in general. The main ingredient for proving the quasi-optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain. Then localization techniques are used to estimate the global L2-error by a local error in the inner part of the domain, where higher regularity for the solution can be assumed, and a global error for the Ritz-projection of the solution, which can be estimated by standard techniques. We validate our results by numerical examples.