Effect of numerical integration for elliptic obstacle problems

Abstract

An elliptic obstacle problem is approximated by piecewise linear finite elements with numerical integration on the penalty and forcing terms. This leads to diagonal nonlinearities and thereby to a practical scheme. Optimal error estimates in the maximum norm are derived. The proof is based on constructing suitable super and subsolutions that exploit the special structure of the penalization, and using quite precise pointwise error estimates for an associated linear elliptic problem with quadrature via the discrete maximum principle.