Stable Discontinuous Galerkin FEM Without Penalty Parameters

John, Lorenz; Neilan, Michael; Smears, Iain

doi:10.1007/978-3-319-39929-4_17

Cited by 11 publications

(14 citation statements)

References 6 publications

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“…is the space composed of homogeneous polynomials of degree k, or (ii) the (larger) polynomial space P k+1 d (T ; R d×d ). For both choices, we prove, using the ideas in [28], that the reconstructed gradient is stable, thereby circumventing the need to introduce and tune any stabilization parameter. Reconstructing the gradient in RTN k d (T ; R d×d ) leads to optimal O(h k+1 )convergence rates for linear problems and smooth solutions, Instead, reconstructing the gradient in P k+1 d (T ; R d×d ) leads to O(h k )-convergence rates for linear problems and smooth solutions, i.e., the method still converges but at a suboptimal order in ideal situations.…”

Section: Introductionmentioning

confidence: 99%

Hybrid High-Order methods for finite deformations of hyperelastic materials

2018

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We devise and evaluate numerically Hybrid High-Order (HHO) methods for hyperelastic materials undergoing finite deformations. The HHO methods use as discrete unknowns piecewise polynomials of order k ≥ 1 on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The discrete problem is written as the minimization of a broken nonlinear elastic energy where a local reconstruction of the displacement gradient is used. Two HHO methods are considered: a stabilized method where the gradient is reconstructed as a tensor-valued polynomial of order k and a stabilization is added to the discrete energy functional, and an unstabilized method which reconstructs a stable higher-order gradient and circumvents the need for stabilization. Both methods satisfy the principle of virtual work locally with equilibrated tractions. We present a numerical study of the two HHO methods on test cases with known solution and on more challenging three-dimensional test cases including finite deformations with strong shear layers and cavitating voids. We assess the computational efficiency of both methods, and we compare our results to those obtained with an industrial software using conforming finite elements and to results from the literature. The two HHO methods exhibit robust behavior in the quasi-incompressible regime.

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Section: Introductionmentioning

confidence: 99%

Hybrid High-Order methods for finite deformations of hyperelastic materials

2018

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“…We consider the choices θ " 1, β " 0, and γ " 0 yielding the symmetric interior penalty method (SIPG) [DD76], θ "´1, β " 0, and γ " 0 which gives the nonsymmetric interior penalty methods (NIPG) [RWG99], and θ " 1, β P R d , and γ " 1 which yields the local discontinuous Galerkin method (LDG) [CS98]; compare also with [ABCM02] and [JNS16]. In all three cases, the corresponding discontinuous Galerkin finite element method (DGFEM) then reads: find u G P VpGq such that…”

Section: The Adgm and The Main Resultsmentioning

confidence: 99%

Convergence of adaptive discontinuous Galerkin methods

Kreuzer

Georgoulis

2018

Math. Comp.

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Abstract. We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables to generalise the basic convergence result for conforming adaptive finite element methods by Morin, Siebert, and Veeser [A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707-737].

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“…Indeed, we can immediately apply a convergence and error estimate result derived from the Gradient Discretisation Method framework. This approach is different from that of [4] where a specific stabilisation term is introduced in the variational formulation, whereas the present work has significant common points with the stable DG method without penalty parameter proposed in [15] (where a scheme, which can enter into the GDM, is proposed, the main difference with the present paper being the lifting of the jumps for computing the discrete gradient).…”

Section: Introductionmentioning

confidence: 83%

“…Let us first state and prove the following lemma, which provides a connection between the discrete gradient defined by (15) to a norm suited for the study of discontinuous Galerkin methods in the framework of elliptic problems.…”

Section: Mathematical Properties Of the Dggd Methodsmentioning

confidence: 99%

Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

Eymard

Guichard

2017

Comp. Appl. Math.

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International audienceWe include in the Gradient Discretisation Method (GDM) framework two numerical schemes based on Discontinuous Galerkin approximations: the Symmetric Interior Penalty Galerkin (SIPG) method, and the scheme obtained by averaging the jumps in the SIPG method. We prove that these schemes meet the main mathematical gradient discretisation properties on any kind of polytopal mesh, by adapting discrete functional analysis properties to our precise geometrical hypotheses. Therefore, these schemes inherit the general convergence properties of the GDM, which hold for instance in the cases of the p−Laplace problem and of the anisotropic and heterogeneous diffusion problem. This is illustrated by simple 1D and 2Dnumerical examples

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Stable Discontinuous Galerkin FEM Without Penalty Parameters

Cited by 11 publications

References 6 publications

Hybrid High-Order methods for finite deformations of hyperelastic materials

Hybrid High-Order methods for finite deformations of hyperelastic materials

Convergence of adaptive discontinuous Galerkin methods

Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

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