# 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…

## 6 Citations

### Stabilised finite element approximation for degenerate convex minimisation problems

- Mathematics
- 2013

Infimising sequences of nonconvex variational problems often do not converge strongly in Sobolev spaces due to fine oscillations. These oscillations are physically meaningful; finite element…

### A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions

- Computer ScienceAdv. Model. Simul. Eng. Sci.
- 2014

Strong convergence of the stress even without any smoothness assumption for a class of stabilised degenerate convex minimisation problems and an improved a posteriori error control is presented, which narrows the efficiency reliability gap.

### Stabilized FEM for Some Optimal Design Problem

- Computer ScienceJ. Sci. Comput.
- 2017

It will be proven that this stabilization technique leads to a posteriori error control on unstructured triangulations, and so enables the use of adaptive algorithms.

### Numerical Algorithms for the Simulation of Finite Plasticity with Microstructures

- Mathematics
- 2015

An adaptive discontinuous Galerkin method for a degenerate convex problem from topology optimization is investigated and some equivalence to nonconforming finite element schemes is established.

### Rate-Independent versus Viscous Evolution of Laminate Microstructures in Finite Crystal Plasticity

- Engineering
- 2015

In this chapter we investigate the variational modeling of the evolution of inelastic microstructures by the example of finite crystal plasticity with one active slip system. For this purpose we…

## References

SHOWING 1-9 OF 9 REFERENCES

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

- Mathematics
- 2007

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

- Mathematics
- 1989

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

- MathematicsClassics 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

- Mathematics
- 2000

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…

### Local Stress Regularity in Scalar Nonconvex Variational Problems

- MathematicsSIAM J. Math. Anal.
- 2002

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

- MathematicsMath. Comput.
- 1997

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