Verifications of primal energy identities for variational problems with obstacles

TL;DR

This paper introduces error identities for obstacle-related free boundary problems, combining energy norms and nonlinear measures, with numerical tests confirming their sharpness and practical relevance.

Contribution

It proposes true error identities for obstacle problems that include nonlinear measures controlling free boundary approximation, advancing error estimation methods.

Findings

Error identities are sharp and reliable.

Nonlinear measures effectively control free boundary approximation.

Different examples show dominance of different error parts.

Abstract

We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm and a certain nonlinear measure. The latter measure controls (in a weak sense) approximation of free boundaries. Numerical tests confirm sharpness of error identities and show that in different examples one or another part of the error measure may be dominant.