Abstract:
Obstacle problems for elliptic and parabolic equations arising in various branches of science and technology are well studied from the mathematical points of view. These studies are mainly focused either on existence of the unique minimizer or on regularity properties of minimizers and respective free boundaries. A systematic overview of these results can be found in the books [1]-[2].
In this talk, we concern with a different question. We study the guaranteed bounds of the difference between the exact solution (minimizer) of the corresponding variational problem and any function (approximation) from the energy class satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle. They can be called estimates of deviations from the exact solution (or a posteriori estimates of functional type). The estimates bound a certain measure (norm) of the error by a functional (error majorant) that depends on the problem data and approximation type, but do not explicitly depend on the exact solution. Hence the functional is fully computable and can be used to evaluate the accuracy of an approximation. Within the framework of this conception, the estimates should be derived on the functional level by the same tools as commonly used in the theory of partial differential equations. They do not use specific features of approximations (e.g., Galerkin orthogonality) what is typical for a posteriori methods applied in mesh adaptive computations based upon finite element technologies. Unlike the a priori rate convergence estimates that establish general asymptotic properties of an approximation method, these a posteriori estimates are applied to a particular solution and allow us to directly verify its accuracy.
We discuss such type estimates for the elliptic thin obstacle problems ([3]), as well as for elliptic biharmonic obstacle problem ([4]). The obtained estimates depend only on the approximate solution (which is known) and on the data of the problem. We emphasise that they also do not need knowledge on the exact coincidence set associated with the exact solution. The obtained error majorants are non-negative and vanishes if and only if the approximation coincides with the exact minimizer. This work is supported by the Russian Science Foundation, grant no. 24-21-00293.
References:
[1] Petrosyan A., Shahgholian H., Uraltseva N. Regularity of Free Boundaries in Obstacle-Type Problems.— Graduate Studies Math., vol. 136. AMS, Providence, Rhode Island, 2012.
[2] Apushkinskaya D. Free Boundary Problems: Regularity Properties Near the Fixed Boundary. — Lecture Notes in Mathematics 2218, Springer, 2018.
[3] Apushkinskaya D. E., Repin S. I. Thin obstacle problem: estimates of the distance to the exact solution, Interfaces Free Bound., 20(4), 511-531 (2018).
[4] Apushkinskaya D. E., Repin S. I. Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions, Comp. Math. Math. Phys., 60(11), 1823–1838 (2020).
