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

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.

Nonuniformly elliptic problems

“…New and interesting counterexamples are in [7,8,9,10]. All this happens in presence of classical uniform ellipticity (2.21), which is somehow counterintuitive according to the narrative drawn at the beginning of this section.…”

Image comparison and scaling via nonlinear elasticity

“…From a rigorous perspective, the numerical method should take into account the possible occurrence of the Lavrentiev phenomenon, whereby the infimum of the energy in A might be strictly less than the infimum among Lipschitz maps in A (such as those generated by a finite-element scheme). For discussions see [6,2,3].…”

Regularity for Double Phase Problems at Nearly Linear Growth

“…Notice that this is essentially the only point where the assumed convexity of t → tg(t) is used (this implies that z → |z|g(|z|) is convex as g(•) is also non-decreasing). For more related results on the absence of Lavrentiev phenomenon we refer to the recent papers [1,3,4,16,50,51] and related references.…”