Adaptive finite elements for obstacle problems

TL;DR

This paper discusses the use of adaptive finite element methods to solve obstacle problems in various applications, addressing challenges in mesh construction and coincidence set resolution.

Contribution

It introduces an adaptive approach using $h$-refinement to effectively handle unknown coincidence sets in obstacle problems across different applications.

Findings

Adaptive $h$-refinement improves solution accuracy

Practical challenges in implementing adaptive methods are identified

Applications include membrane contact, elastoplastic torsion, and cavitation modeling

Abstract

We summarise three applications of the obstacle problem to membrane contact, elastoplastic torsion and cavitation modelling, and show how the resulting models can be solved using mixed finite elements. It is challenging to construct fixed computational meshes for any inequality-constrained problem because the coincidence set has an unknown shape. Consequently, we demonstrate how $h$-adaptivity can be used to resolve the unknown coincidence set. We demonstrate some practical challenges that must be overcome in the application of the adaptive method.