A hybridizable discontinuous Galerkin method for linear elasticity

Soon, See-Chew; Cockburn, Bernardo; Stolarski, Henryk K.

doi:10.1002/nme.2646

Cited by 113 publications

(160 citation statements)

References 31 publications

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“…In our setting, this can be done by taking the LDG method to define the discrete local solvers and the hybrid unknown λ in the space of continuous piecewise continuous functions. However, in [25] it was proven that the resulting method, which was called the embedded discontinuous Galerkin method (EDG) [45], loses the convergence properties of the associated HDG method and behaves closely to that of the CG method.…”

Section: Discussionmentioning

confidence: 99%

“…A similar effect is produced when instead of taking λ in the space of discontinuous functions, it is taken in a subspace of continuous functions; see [25]. The resulting methods, introduced in [45], were called the embedded discontinuous Galerkin (EDG) methods, and have a stiffness matrix for λ with a sparsity structure identical to that of the statically condensed CG method. A similar idea prompted the introduction of the so-called multi-scale discontinuous Galerkin (MDG) method [10,36].…”

Section: Introductionmentioning

confidence: 99%

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To CG or to HDG: A Comparative Study

2011

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Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle hp-adaptivity.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

To CG or to HDG: A Comparative Study

2011

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“…The HDG method has proven to be a popular method and has, in recent years, been applied in the context of steady-state di↵usion [9,10,16,18], Maxwell's equations [32,33,36], convection-di↵usion problems [7,13,34,35], linear elasticity [45], Timoshenko beam model [4,5], elastodynamics [37], Stokes equations [12,17,19,34], compressible [26,49] and incompressible Navier-Stokes, and Oseen equations [6,37,39,40]. We note, however, that the majority of these works focus either on the theoretical aspects of the method such as formulation and analysis for a specific equation type, or the specific benefits such as accurately captured solution features that the HDG method can o↵er.…”

Section: Introductionmentioning

confidence: 99%

To CG or to HDG: A Comparative Study in 3D

et al. 2015

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Since the inception of discontinuous Galerkin (DG) methods for elliptic problems, there has existed a question of whether DG methods can be made more computationally e cient than continuous Galerkin (CG) methods. Fewer degrees of freedom, approximation properties for elliptic problems together with the number of optimization techniques, such as static condensation, available within CG framework make it challenging for DG methods to be competitive until recently. However, with the introduction of a static-condensation-amenable DG method -the hybridizable discontinuous Galerkin (HDG) method -it has become possible to perform a realistic comparison of CG and HDG methods when applied to elliptic problems. In this work, we extend upon an earlier 2D comparative study, providing numerical results and discussion of the CG and HDG method performance in three dimensions. The comparison categories covered include steady-state elliptic and time-dependent parabolic problems, various element types and serial and parallel performance. The postprocessing technique, which allows for superconvergence in the HDG case, is also discussed. Depending on the linear system solver used and the type of the problem (steady-state vs time-dependent) in question the HDG method either outperforms or demonstrates a comparable performance when compared with the CG method. The HDG method however falls behind performance-wise when the iterative solver is used, which indicates the need for an e↵ective preconditioning strategy for the method.

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“…Robustness, local conservation and flexibility for implementing h and padaptivity strategies are well known advantages of DG methods stemming from the use of finite element spaces consisting of discontinuous piecewise polynomials. A natural connection between DG formulations and hybrid methods have been exploited successfully in many problems [11][12][13][14][15] and they are still being developed. These hybrid formulations have improved stability, robustness and flexibility of the DG methods with reduced complexity and computational cost.…”

Section: Introductionmentioning

confidence: 99%

A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

Santos

Faria

Loula

2018

Tend. Mat. Apl. Comput.

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ABSTRACT. In this work, a primal hybrid finite element method for nearly incompressible linear elasticity problem on triangular meshes is shown. This method consists of coupling local discontinuous Galerkin problems to the primal variable with a global problem for the Lagrange multiplier, which is identified as the trace of the displacement field. Also, a local post-processing technique is used to recover stress approximations with improved rates of convergence in H(div) norm. Numerical studies show that the method is locking free even using equal or different orders for displacement and stress fields and optimal convergence rates are obtained.

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A hybridizable discontinuous Galerkin method for linear elasticity

Cited by 113 publications

References 31 publications

To CG or to HDG: A Comparative Study

To CG or to HDG: A Comparative Study

To CG or to HDG: A Comparative Study in 3D

A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

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