A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

Gudi, Thirupathi; Porwal, Kamana

doi:10.1016/j.cam.2015.07.008

Cited by 22 publications

(15 citation statements)

References 47 publications

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“…We recall that the discrete variational inequality (6) for which we derived the a posteriori estimator in Section 3.2 is part of the discrete phase-field model (Problem 2). Due to the equivalence of Problem 2 and the complementarity system (15) it holds that Λ h , φ p −1,1 in (15) and (7) are the same. The only difference is that now we have chosen a discrete basis for Λ h such that the node value…”

Section: Solution Algorithmsmentioning

confidence: 99%

“…An extension to a functional on H 1 was proposed by means of lumping

\sum_{p \in {frakturN}^{C}} s_{p} φ_{p}

, where

\begin{matrix} alignleft \\ align-1 s_{p} = \frac{{⟨ Λ_{h}, φ_{p} ⟩}_{- 1, 1}}{\int_{ω_{p}} φ_{p} d x} \geq 0 & align-2 \end{matrix}

are the node values of the lumped discrete constraining force. This approach has been extended and applied to different obstacle and contact problems …”

Section: Discretization and A Posteriori Error Estimatormentioning

confidence: 99%

“…By measuring the error in the solutions as well as in the constraining force, the first efficient and reliable residual‐type estimator for obstacle problems has been derived . This approach has been extended to discontinuous Galerkin methods by Gudi and Porwal . Error estimators for contact problems are given by Hild and Nicaise, Weiss and Wohlmuth, and Krause et al The local structure of the solution and constraining force has been exploited to localize the estimator contribution related to the constraints .…”

Section: Introductionmentioning

confidence: 99%

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Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

et al. 2019

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In this work, we consider adaptive mesh refinement for a monolithic phase‐field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase‐field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual‐type error estimator for the phase‐field fracture model in each time‐step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle‐point system has three unknowns: displacements, phase‐field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.

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Section: Solution Algorithmsmentioning

confidence: 99%

“…An extension to a functional on H 1 was proposed by means of lumping

\sum_{p \in {frakturN}^{C}} s_{p} φ_{p}

, where

\begin{matrix} alignleft \\ align-1 s_{p} = \frac{{⟨ Λ_{h}, φ_{p} ⟩}_{- 1, 1}}{\int_{ω_{p}} φ_{p} d x} \geq 0 & align-2 \end{matrix}

are the node values of the lumped discrete constraining force. This approach has been extended and applied to different obstacle and contact problems …”

Section: Discretization and A Posteriori Error Estimatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

et al. 2019

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“…Recently, DG methods have been applied for solving VIs, such as gradient plasticity problem [27,28], obstacle problems [29,30], Signorini problem [31,32], quasistatic contact problems [33], plate contact problem [34][35][36], two membranes problem [37] and Stokes or Navier-Stokes flows with slip boundary condition [38,39]. A posteriori error analysis of DG methods for VIs was also considered in [40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin methods for solving a hyperbolic inequality

Wang

Han

2018

Numerical Methods Partial

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In this paper, we study spatially semi‐discrete and fully discrete schemes to numerically solve a hyperbolic variational inequality, with discontinuous Galerkin (DG) discretization in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for four DG schemes, which reaches the optimal convergence order for linear elements. A numerical example is presented, and the numerical results confirm the theoretical error estimates.

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“…Numerous references can be found on the numerical approximation and their general convergence analysis of variational inequalities, see for example, [5][6][7][8][9][10] for the work on obstacle problem and refer to for example, [2,7,[11][12][13][14] for the analysis of Signorini problem. The work related to a posteriori analysis of finite element methods for variational inequalities has begun not too long ago, for example, refer to [15][16][17][18][19][20][21][22][23][24][25][26][27] for a posteriori error analysis of variational inequalities of the first kind.…”

Section: Introductionmentioning

confidence: 99%

A C⁰ interior penalty method for a fourth‐order variational inequality of the second kind

Gudi

Porwal

2015

Numerical Methods Partial

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In this article, we propose a C0 interior penalty ( C 0 I P ) method for the frictional plate contact problem and derive both a priori and a posteriori error estimates. We derive an abstract error estimate in the energy norm without additional regularity assumption on the exact solution. The a priori error estimate is of optimal order whenever the solution is regular. Further, we derive a reliable and efficient a posteriori error estimator. Numerical experiments are presented to illustrate the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 36–59, 2016

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A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

Cited by 22 publications

References 47 publications

Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

Discontinuous Galerkin methods for solving a hyperbolic inequality

A C⁰ interior penalty method for a fourth‐order variational inequality of the second kind

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A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

Cited by 22 publications

References 47 publications

Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

Discontinuous Galerkin methods for solving a hyperbolic inequality

A C0 interior penalty method for a fourth‐order variational inequality of the second kind

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A C⁰ interior penalty method for a fourth‐order variational inequality of the second kind