Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem
For researchers in numerical analysis and computational mechanics, this work provides rigorous error bounds for a common regularization approach to the obstacle problem, though the assumption of inactive obstacle on the boundary limits its generality.
This paper proves quasi-optimal finite element error estimates in the L^2-norm for the obstacle problem using a regularized discrete approximation, achieving optimal convergence rates under the assumption that the obstacle is inactive on the boundary. Numerical examples confirm the theoretical results.
This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-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 $L^2$-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.