TITLE

# Î“-limits and relaxations for rate-independent evolutionary problems

AUTHOR(S)
Mielke, Alexander; Roubíček, Tomáš; Stefanelli, Ulisse
PUB. DATE
March 2008
SOURCE
Calculus of Variations & Partial Differential Equations;Mar2008, Vol. 31 Issue 3, p387
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals $${\mathcal{E}}$$ and the dissipation distance $${\mathcal{D}}$$ . For sequences $$({\mathcal{E}}_k)_{k\in {\mathbb{N}}}$$ and $$({\mathcal{D}}_k)_{k\in {\mathbb{N}}}$$ we address the question under which conditions the limits q âˆž of solutions $$q_k : [0, T]\to {\mathcal{Q}}$$ satisfy a suitable limit problem with limit functionals $${\mathcal{E}}_\infty$$ and $${\mathcal{D}}_\infty$$ , which are the corresponding Î“-limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q âˆž solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k â†’ âˆž, which in the limit can be described by an effective macroscopic model.
ACCESSION #
27960664

## Related Articles

• Numerical approximation of gradient flows for closed curves in â„d. BARRETT, JOHN W.; GARCKE, HARALD; NÜRNBERG, ROBERT // IMA Journal of Numerical Analysis;Jan2010, Vol. 30 Issue 1, p4

We present parametric finite-element approximations of curvature flows for curves in â„d, where d â‰¥ 2, as well as for curves on two-dimensional manifolds in â„3. Here we consider the curve shortening flow, the curve diffusion and the elastic flow. It is demonstrated that the...

• Dual-primal FETI algorithms for edge finite-element approximations in 3D. Toselli, Andrea // IMA Journal of Numerical Analysis;Jan2006, Vol. 26 Issue 1, p96

A family of dual-primal finite-element tearing and interconnecting methods for edge-element approximations in 3D is proposed and analysed. The key part of this work relies on the observation that for these finite-element spaces there is a strong coupling between degrees of freedom associated...

• Gradient Recovery for Singularly Perturbed Boundary Value Problems I: One-Dimensional Convection-Diffusion. Roos, H.-G.; Lin?, T. // Computing;2001, Vol. 66 Issue 2, p163

We consider a Galerkin finite element method that uses piecewise linears on a class of Shishkin-type meshes for a model singularly perturbed convection-diffusion problem. We pursue two approaches in constructing superconvergent approximations of the gradient. The first approach uses...

• A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem***. Benhassine, Hani; Bendali, Abderrahmane // ESAIM: Mathematical Modelling & Numerical Analysis;Jul2011, Vol. 45 Issue 4, p675

This study is mainly dedicated to the development and analysis of non-overlapping domain decomposition methods for solving continuous-pressure finite element formulations of the Stokes problem. These methods have the following special features. By keeping the equations and unknowns unchanged at...

• Approximation of functions with growing derivatives. Arsent'eva, E.; Dem'yanovich, Yu. // Journal of Mathematical Sciences;Nov2011, Vol. 178 Issue 6, p565

We consider nondegenerate (simplicial) subdivisions (triangulation), refined near the boundary of a domain, and derive weight estimates for the Courant approximation of functions whose second order derivatives grow (near the boundary). Bibliography: 7 titles.

• Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations. Li, Qin; Lin, Qun; Xie, Hehu // Applications of Mathematics;Apr2013, Vol. 58 Issue 2, p129

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q, EQ and enriched Crouzeix-Raviart. We first...

• Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers. Jichun Li; Yitung Chen // Numerical Mathematics: Theory, Methods & Applications;May2008, Vol. 1 Issue 2, p138

In this paper, using Lin's integral identity technique, we prove the optimal uniform convergence í’ª(Nx-2lnÂ²Nx + Ny-2lnÂ²Ny) in the LÂ²-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh....

• Superconvergence Analysis of Finite Element Method for a Second-Type Variational Inequality. Dongyang Shi; Hongbo Guan; Xiaofei Guan // Journal of Applied Mathematics;2012, p1

This paper studies the finite element (FE) approximation to a second-type variational inequality. The supe rclose and superconvergence results are obtained for conforming bilinear FE and nonconforming EQrot FE schemes under a reasonable regularity of the exact solution u âˆŠ H5/2(O), which...

• A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes. Brenner, Andreas; Bänsch, Eberhard; Bause, Markus // IMA Journal of Numerical Analysis;Jan2014, Vol. 34 Issue 1, p123

In this article we study finite element approximations of the time-dependent Stokes system on dynamically changing meshes. Applying the backward Euler method for time discretization we use the discrete Helmholtz or Stokes projection to evaluate the solution at time tnâˆ’1 on the new spatial...

Share