Nonconforming finite-element discretization of convex variational problems

Abstract

[Received on ]The Lavrentiev gap phenomenon is a well-known effect in the calculus of variations, related to singularities of minimizers. In its presence, conforming finite element methods are incapable of reaching the energy minimum. By contrast, it is shown in this work that, for convex variational problems, the non-conforming Crouzeix-Raviart finite element discretization always converges to the correct minimizer, and that the discrete energy converges to the correct limit.

Analysis of a Class of Penalty Methods for Computing Singular Minimizers

“…However, at this point it stagnates and is unable to lower the penalty parameter further without increasing the energy above E goal . This is strong indication that no Lavrentiev gap exists or, more precisely, that no gap larger than 10 −3 exists, which is consistent with [25,26]. Next, we considered the case ν = 1/40 and p = 4.…”

“…We choose ν = 1/60 and p = 6 a case for which numerical experiments in [25,26] indicate the absence of a Lavrentiev gap. An adaptive Galerkin solution suggests that the infimum of the energy in the space of Lipschitz functions is approximately inf E(V ∩ W 1,∞ (Ω; R 2 )) ≈ 0.0093 + O(3 × 10 4 ).…”

“…In [25,26] non-conforming finite element methods were analyzed as an alternative to the penalty methods discussed in the present paper. The main advantage of non-conforming methods is that they require no penalty parameter.…”

“…The main advantage of non-conforming methods is that they require no penalty parameter. However, even though this is a promising new direction, it is at present entirely unclear how to generalize the results in [25,26] to the vectorial non-convex case. By contrast, our convergence results in the present paper hold under far less restrictive conditions on the stored energy functions.…”

The nonconforming virtual element method

“…In the case of second order elliptic problems, the approximation space has some continuity built in it, but still discrete functions are not continuous. Still, that relaxed continuity (or crime) has proved its usefulness in many applications, mostly related to continuum mechanics, in particular, for fluid flow problems [26,34] (for moderate Reynolds numbers) and elasticity [22,32].…”