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$Γ$-convergence of an Enhanced Finite Element Method for Manià's and Foss's Problems Exhibiting the Lavrentiev Gap Phenomenon

Xiaobing Feng, Joshua M. Siktar

TL;DR

This work addresses the Lavrentiev Gap Phenomenon that defeats standard finite element methods for Manià's one-dimensional variational problem and Foss's two-dimensional elasticity problem. It develops an enhanced finite element framework using a cutoff function to define modified discrete functionals and proves Γ-convergence to the original functionals in fractional Sobolev spaces, via recovery sequences built from a stable interpolation operator. The results include complete Γ-convergence proofs for both problems, together with convergence of minimizers, and are complemented by numerical experiments that corroborate convergence behavior under limited regularity. The methods provide a principled blueprint for discretizing LGP-affected variational problems and point to future extensions to other LGP contexts, including higher dimensions and nonlocal formulations.

Abstract

It is well-known that numerically approximating calculus of variations problems possessing a Lavrentiev Gap Phenomenon (LGP) is challenging, and the standard numerical methodologies, such as finite element, finite difference, and discontinuous Galerkin methods, fail to give convergent methods because they cannot overcome the gap. This paper is a continuation of a 2016 paper by Feng and Schnake, where a promising enhanced finite element method was proposed to overcome the LGP in the classical Manià's problem. The first goal of this paper is to provide a complete $Γ$-convergence proof for this enhanced finite element method, hence, establishing a theoretical foundation for the method. The crux of the convergence analysis is taking advantage of the regularity of the minimizer and viewing the minimization problem as posed over the fractional Sobolev space $W^{1 + s, p}(0, 1)$ (for $s > 0$) rather than the original admissible space $W^{1, p}(0, 1)$. The second goal is to extend the enhanced finite element method to the two-dimensional Foss's problem from nonlinear elasticity, which is also known to possess the LGP, and to establish its $Γ$-convergence as well.

$Γ$-convergence of an Enhanced Finite Element Method for Manià's and Foss's Problems Exhibiting the Lavrentiev Gap Phenomenon

TL;DR

This work addresses the Lavrentiev Gap Phenomenon that defeats standard finite element methods for Manià's one-dimensional variational problem and Foss's two-dimensional elasticity problem. It develops an enhanced finite element framework using a cutoff function to define modified discrete functionals and proves Γ-convergence to the original functionals in fractional Sobolev spaces, via recovery sequences built from a stable interpolation operator. The results include complete Γ-convergence proofs for both problems, together with convergence of minimizers, and are complemented by numerical experiments that corroborate convergence behavior under limited regularity. The methods provide a principled blueprint for discretizing LGP-affected variational problems and point to future extensions to other LGP contexts, including higher dimensions and nonlocal formulations.

Abstract

It is well-known that numerically approximating calculus of variations problems possessing a Lavrentiev Gap Phenomenon (LGP) is challenging, and the standard numerical methodologies, such as finite element, finite difference, and discontinuous Galerkin methods, fail to give convergent methods because they cannot overcome the gap. This paper is a continuation of a 2016 paper by Feng and Schnake, where a promising enhanced finite element method was proposed to overcome the LGP in the classical Manià's problem. The first goal of this paper is to provide a complete -convergence proof for this enhanced finite element method, hence, establishing a theoretical foundation for the method. The crux of the convergence analysis is taking advantage of the regularity of the minimizer and viewing the minimization problem as posed over the fractional Sobolev space (for ) rather than the original admissible space . The second goal is to extend the enhanced finite element method to the two-dimensional Foss's problem from nonlinear elasticity, which is also known to possess the LGP, and to establish its -convergence as well.
Paper Structure (12 sections, 14 theorems, 54 equations, 8 tables)

This paper contains 12 sections, 14 theorems, 54 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Facts about Manià's and Foss's problems
  5. Enhanced finite element methods and fractional inverse inequalities
  6. Finite element spaces and interpolations
  7. Enhanced finite element methods
  8. Fractional inverse inequalities
  9. $\Gamma$-convergence for Manià's problem
  10. $\Gamma$-convergence for Foss's problem
  11. Numerical experiments
  12. Conclusion

Key Result

Lemma 3.1

\newlabellemma3.1 Let $\Omega=(0,1)$ or $\Omega=(0, 1) \times \left(\frac{3}{2}, \frac{5}{2}\right)$ and $v \in W^{r, p}(\Omega)$ for $0 < r \leq 2$ and $1 \leq p < \infty$, then $I_h$ satisfies the following properties: If we write $r=1+s$ for $0< s\leq 1$, the right-hand side of property-3 becomes $h^s[v]_{W^{1 + s, p}(\Omega)}$.

Theorems & Definitions (29)

  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.2: Fractional inverse inequality, $p = 2$
  • Theorem 3.3: Generalized fractional inverse inequality
  • proof
  • Corollary 3.4
  • Definition 4.1: $\Gamma$-convergence
  • Lemma 4.2
  • proof
  • ...and 19 more