First-Order Least-Squares Method for the Obstacle Problem
This work offers a new numerical method for solving obstacle problems, which are important in contact mechanics and fluid dynamics, but the contribution is incremental as it extends existing least-squares techniques to a specific problem class.
The paper develops a least-squares finite element method for the obstacle problem, providing a priori error estimates with optimal convergence rates and a posteriori error bounds for adaptive algorithms, validated by numerical studies.
We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide also a posteriori bounds that can be be used as error indicators in an adaptive algorithm. Numerical studies are presented.