On the Strong Convergence of Gradients in Stabilized Degenerate Convex Minimization Problems

@article{Boiger2010OnTS,
  title={On the Strong Convergence of Gradients in Stabilized Degenerate Convex Minimization Problems},
  author={Wolfgang Boiger and Carsten Carstensen},
  journal={SIAM J. Numer. Anal.},
  year={2010},
  volume={47},
  pages={4569-4580}
}
Infimizing sequences in nonconvex variational problems typically exhibit enforced finer and finer oscillations called microstructures such that the infimal energy is not attained. Although those oscillations are physically meaningful, finite element approximations experience difficulty in their reconstruction. The relaxation of the nonconvex minimization problem by (semi) convexification leads to a macroscopic model for the effective energy. The resulting discrete macroscopic problem is… 

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References

SHOWING 1-9 OF 9 REFERENCES

Convergence of adaptive FEM for a class of degenerate convex minimization problems

A class of degenerate convex minimization problems allows for some adaptive finite-element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of

Direct methods in the calculus of variations

Introduction.- Convex Analysis and the Scalar Case.- Convex Sets and Convex Functions.- Lower Semicontinuity and Existence Theorems.- The one Dimensional Case.- Quasiconvex Analysis and the Vectorial

The finite element method for elliptic problems

  • P. Ciarlet
  • Mathematics
    Classics in applied mathematics
  • 2002
From the Publisher: This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional

Elements of Nonlinear Analysis

1. Some Physical Motivations.- 1.1. An elementary theory of elasticity.- 1.2. A problem in biology.- 1.3. Exercises.- 2. A Short Background in Functional Analysis.- 2.1. An introduction to

Variational models for microstructure and phase transitions

Local Stress Regularity in Scalar Nonconvex Variational Problems

This paper addresses convex but not necessarily strictly convex minimization problems, and shows regularity up to the boundary in a class of energy functionals for which any stress field $\sigma$ in $L^q(\Omega)$ with $\operatorname{{\rm div}}\sigma $ belongs to $ W^{1,q}_{loc}(\ Omega)$.

Numerical solution of the scalar double-well problem allowing microstructure

This work treats the scalar double-well problem by numerical solution of the relaxed problem (RP) leading to a (degenerate) convex minimisation problem and proves a priori and a posteriori estimates for σ-σ h in L 4/3 (Ω) and weaker weighted estimates for ⊇u - ⊽u h .

Convergence for stabilisation of degenerate convex minimsation problems

  • Interfaces Free Bound., 6
  • 2004