A Truncation Method for Detecting Singular Minimizers Involving the Lavrentiev Phenomenon

Abstract

Abstract. A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on the Manià's example and a 2-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent dependence, are given to show the efficiency of the method.

Numerical Solution of Nonlinear Elasticity Problems With Lavrentiev Phenomenon

“…In [2,18], the convergence results of the truncation method for the case when f is convex were obtained for some specially designed truncation functions. With similar techniques as used in [2], we establish below the convergence theorems of the truncation method for the case when f is polyconvex.…”

“…In the present paper, we generalize the theory developed in [2] and establish a convergence theory for the truncation method for the case when f is polyconvex, which enables reliable applications of the truncation method to polyconvex elasticity problems.…”

“…Various numerical methods for detecting singular minimizers have been developed in recent years [2,6,16,17,18,19] (see [7] for a survey and more references), and the corresponding convergence theorems were proved for the case when the integrand f is convex with respect to the deformation gradient Du.…”

Nonconforming finite-element discretization of convex variational problems

“…In this way one obtains convergence results of the following type: for any sequence ε j 0 there exists a sequence h j 0 such that minimizers of J εj in A ∩ P c 1 (T hj ) m converge weakly to a minimizer u of J in A 1 , and J εj (u hj ) → J (u). Methods of this type include the penalty method of Ball and Knowles [5,18] and its extension to polyconvex integrands by Negron-Marrero [23], the element-removal method of Li [20,21], the truncation method of Bai and Li [2], and the L 1 -penalty method of Carstensen and Ortner [11]. The main advantage of these methods is their generality, as they are in principle appropriate for very general classes of minimization problems.…”

Analysis of a Class of Penalty Methods for Computing Singular Minimizers

“…Several classes of numerical schemes have been introduced in the literature to allow for a numerical detection of (L), including the penalty method of Ball and Knowles [5,17] and its extension to polyconvex integrands by Negron-Marrero [24], the element-removal method of Li [19,20], and the truncation method of Li, and Bai and Li [1,2,21].…”