An adaptive finite element method for the inequality-constrained Reynolds equation

TL;DR

This paper introduces a stabilized finite element method for solving the inequality-constrained Reynolds equation in lubrication, providing optimal error estimates and demonstrating improved numerical performance over traditional penalty methods.

Contribution

It develops a residual-based stabilized finite element approach with novel a posteriori error estimators for cavitation modeling in lubrication.

Findings

The method achieves optimal a priori error estimates.

Numerical results show superiority over penalty methods.

The approach effectively handles inequality constraints in lubrication problems.

Abstract

We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem and approximated by a residual-based stabilized method. Based on our recent results on the classical obstacle problem, we present optimal a priori estimates and derive novel a posteriori error estimators. The method is implemented as a Nitsche-type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods.