Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap

TL;DR

This paper demonstrates that the Crouzeix-Raviart finite element method converges for non-autonomous variational problems with Lavrentiev gaps, where conforming methods fail, supported by theoretical proofs and numerical experiments.

Contribution

It proves convergence of the Crouzeix-Raviart scheme for problems with Lavrentiev gaps, a case where conforming methods do not work, providing new analytical and numerical insights.

Findings

Crouzeix-Raviart method converges despite Lavrentiev gap

Conforming schemes fail in these problems

Numerical experiments support theoretical results

Abstract

We investigate the convergence of the Crouzeix-Raviart finite element method for variational problems with non-autonomous integrands that exhibit non-standard growth conditions. While conforming schemes fail due to the Lavrentiev gap phenomenon, we prove that the solution of the Crouzeix-Raviart scheme converges to a global minimiser. Numerical experiments illustrate the performance of the scheme and give additional analytical insights.