A FEM approximation of a two-phase obstacle problem and its a posteriori error estimate

TL;DR

This paper develops a finite element approximation method for the two-phase obstacle problem, utilizing a dual formulation and a posteriori error estimates to ensure accuracy and identify free boundaries.

Contribution

It introduces a novel dual-based finite element algorithm for the two-phase obstacle problem with guaranteed error bounds and free boundary indicators.

Findings

Numerical verification of basic estimates on two examples.

The dual problem reduces to a quadratic program with box constraints.

The a posteriori error estimate provides a guaranteed upper bound.

Abstract

This paper is concerned with the two--phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of Repin and Valdman (2015) and verify them numerically on two examples in two space dimensions. A solution algorithm is proposed for the construction of the finite element approximation to the two--phase obstacle problem. The algorithm is not based on the primal (convex and nondifferentiable) energy minimization problem but on a dual maximization problem formulated for Lagrange multipliers. The dual problem is equivalent to a quadratic programming problem with box constraints. The quality of approximations is measured by a functional a posteriori error estimate which provides a guaranteed upper bound of the difference of approximated and exact energies of the primal minimization problem. The majorant functional in the upper bound contains auxiliary variables and it is optimized with respect to them to provide a sharp upper bound. A space density of the nonlinear related part of the majorant functional serves as an indicator of the free boundary.