Verification of functional a posteriori error estimates for obstacle problem in 1D

TL;DR

This paper verifies a functional a posteriori error estimate for the 1D obstacle problem by deriving an exact solution, constructing a benchmark, and comparing numerical approximations with the exact solution to assess error bounds.

Contribution

It simplifies the obstacle problem to 1D, constructs an exact solution benchmark, and evaluates the sharpness of the a posteriori error estimate for finite element approximations.

Findings

The a posteriori error estimate provides a reliable upper bound.

The constructed benchmark confirms the sharpness of the error estimate.

Finite element solutions closely match the exact solution within the estimated bounds.

Abstract

We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.