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.

On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

“…The absence of (3) does not necessary lead to a discrepancy between W and H, indeed, other sufficient conditions for the equality W = H exist and the counterexamples to this equality are quite scarce (see, however, [15] for recent results in this direction). Following [52] (see also [2,15] for variants of the example; the precise formulation we take involves a rotation, like in [16]), in the square (−1, 1) 2 of R 2 consider the piecewise constant variable exponent with a saddle-point at the origin:…”

“…For this choice of the domain and of the variable exponent, the above definitions lead to W \ H = ∅, moreover, H is of co-dimension one in W (see [2,16] and references therein).…”

“…In the present note, we combine the insight from these results and from the recent work [16] with the standard finite volume discretization framework(s) for the PDEs of p-laplacian kind. As the common guidelines of [16] and of our investigation, one can highlight the following properties:…”

“…Poincaré-Friedrichs inequality holds true). According to the choice of the numerical method, convergence to the narrow or to the broad solution can be witnessed; we refer in particular to the recent work [16] in the variational setting and for finite element methods. We also refer to [23] for a priori error analysis of finite element approximations of p(x)-laplace problems.…”

Convergent adaptive hybrid higher-order schemes for convex minimization

“…Its presence is equivalent to the failure of standard conforming FEMs [17, Theorem 2.1] in the sense that a wrong minimal energy is approximated. As a remedy, the nonconforming Crouzeix-Raviart FEM in [51,52,3] can overcome the Lavrentiev gap under fairly general assumptions on W : Throughout the remaining parts of this section, let W ∈ C 1 (M) be convex with the one-sided lower growth c 1 |A| p − c 2 ≤ W (A) for all A ∈ M and some 1 < p < ∞.…”