Deviation identity for linear differential operators and its application to obstacle problems
Kseniya Darovskaya
TL;DR
The paper presents a deviation identity for linear variational problems with a coercive operator, linking the primal and dual errors to a computable residual and enabling a posteriori error assessment that is independent of how approximations are constructed. It first establishes a general equality relating $\|\Lambda(v-u)\|_{\mathcal{A}}$, $\|p^*-y^*\|_{\mathcal{A}^{-1}}$, and a computable residual, then specializes to a biharmonic obstacle problem. In the biharmonic case, it derives an explicit decomposition $M_{\mathbb K}=\mu_\varphi(v)+\mu^*_{\varphi}(y^*)$ and obtains the a posteriori identity with a sign condition $f-\operatorname{divDiv} y^* \le 0$ ensuring nonnegativity, providing a practical tool for estimating deviations from the exact solution in obstacle-type variational problems. The work connects to prior a posteriori results and notes potential applicability to ML-PDE solvers and generic variational problems beyond the biharmonic setting.
Abstract
For a class of variational problems with linear differential operator, we obtain a convenient form of the deviation identity, i.e., the value of the distance between approximated solutions and the exact ones. We illustrate the result with an explicit form of the deviation identity for a biharmonic obstacle problem.