Error Estimates for Space-Time Discretizations of a Rate-Independent Variational Inequality

Mielke, Alexander; Paoli, Lætitia; Petrov, Adrien; Stefanelli, Ulisse

doi:10.1137/090750238

Cited by 27 publications

(20 citation statements)

References 49 publications

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“…This highlights that the assumed (non-)convexity has to be appropriate in order for one to be able to show the existence, uniqueness and regularity of solutions. It is for this reason that the existence, uniqueness and regularity theory developed in this paper remains valid up until the moment when the solution reaches a point of non-convexity; see Remark 3.2. Partial results in the direction of regularity have already been presented in [13,16,23,26], but those papers deal with the Mielke-Theil energetic solution concept. Here, we are able to guarantee more regularity in both time and space, and with the natural dependence of the regularity of the solution on that of the data.…”

Section: Introductionmentioning

confidence: 99%

Regularity and approximation of strong solutions to rate-independent systems

Rindler

Schwarzacher

Süli

2017

Math. Models Methods Appl. Sci.

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Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this work we prove the existence of Hölder-regular strong solutions for a class of rate-independent systems. We also establish additional higher regularity results that guarantee the uniqueness of strong solutions. The proof proceeds via a time-discrete Rothe approximation and careful elliptic regularity estimates depending in a quantitative way on the (local) convexity of the potential featuring in the model. In the second part of the paper we show that our strong solutions may be approximated by a fully discrete numerical scheme based on a spatial finite element discretization, whose rate of convergence is consistent with the regularity of strong solutions whose existence and uniqueness are established.MSC (2010): 49J40 (primary); 47J20, 47J40, 74H30.

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Section: Introductionmentioning

confidence: 99%

Regularity and approximation of strong solutions to rate-independent systems

Rindler

Schwarzacher

Süli

2017

Math. Models Methods Appl. Sci.

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“…In the same way as in , the following estimate is valid Observe that . Now, arguments similar to – show that and the proof of Theorem is finished. Example Energies of the type occur for instance in the Souza–Auricchio model describing shape memory alloys. Let

u MathClass-punc: Ω MathClass-rel\to R^{d}

denote the displacement field and the transformation strain.…”

Section: Examples and Extensionsmentioning

confidence: 99%

“…In the rate independent framework, for shape‐memory alloys, the dissipation potential is typically given by

scriptR MathClass-open(z MathClass-close) MathClass-rel= \int_{Ω} ρ MathClass-open(x MathClass-close) (||, z) normald x

for some L ∞ ‐coefficient ρ ≥ ρ 0 > 0. The evolution law is formulated in terms of a global stability criterion based on stable sets

scriptS MathClass-open(t MathClass-close)

and an energy balance. Here, for fixed time t , the stable set

scriptS MathClass-open(t MathClass-close)

is defined as

scriptS MathClass-open(t MathClass-close) MathClass-rel= MathClass-open{MathClass-open(u MathClass-punc, z MathClass-close) MathClass-rel\in H_{Γ_{D}}^{1} MathClass-open(Ω MathClass-close) MathClass-bin\times H^{1} MathClass-open(Ω MathClass-close) MathClass-punc; scriptW MathClass-open(u MathClass-punc, z MathClass-close) MathClass-rel\leq scriptW MathClass-open(v MathClass-punc, ζ MathClass-close) MathClass-bin+ scriptR MathClass-open(z MathClass-bin- ζ MathClass-close) MathClass-punc, MathClass-open(v MathClass-punc, ζ MathClass-close) MathClass-rel\in H_{Γ_{D}}^{1} MathClass-open(Ω MathClass-close) MathClass-bin\times H^{1} MathClass-open(Ω MathClass-close) MathClass-close}

.…”

Section: Examples and Extensionsmentioning

confidence: 99%

“…Energies of the type (3.9) occur for instance in the Souza-Auricchio model describing shape memory alloys [22,23]. Let u : !…”

Section: Example 39mentioning

confidence: 99%

“…In the rate independent framework, for shape-memory alloys, the dissipation potential is typically given by R.z/ D R .x/ jzj dx for some L 1 -coefficient 0 > 0. The evolution law is formulated in terms of a global stability criterion based on stable sets S.t/ and an energy balance [21,23]. Here, for fixed time t, the stable set S.t/ is defined as…”

Section: Example 39mentioning

confidence: 99%

See 2 more Smart Citations

Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Knees

Schröder

2012

Math Methods in App Sciences

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A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D), the displacement fields are regular. The proof relies on a difference quotient argument for the directions tangential to the crack. To obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann (2002) is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. On the basis of Falk's approximation theorem for variational inequalities, convergence rates for finite element discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.

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Rate-independent systems in Banach spaces

Mielke

Roubíček

2015

Applied Mathematical Sciences

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Error Estimates for Space-Time Discretizations of a Rate-Independent Variational Inequality

Cited by 27 publications

References 49 publications

Regularity and approximation of strong solutions to rate-independent systems

Regularity and approximation of strong solutions to rate-independent systems

Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Rate-independent systems in Banach spaces

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