Low-order dPG-FEM for an elliptic PDE

Carstensen, Carsten; Gallistl, Dietmar; Hellwig, Friederike; Weggler, Lucy

doi:10.1016/j.camwa.2014.09.013

Cited by 22 publications

(18 citation statements)

References 6 publications

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“…Now, further investigations on the performance of the discontinuous Petrov-Galerkin discretization method can be accomplished. The convergence during uniform and adaptive mesh refinement for a lowest-order dPG FEM for the Poisson Model Problem is given in [4] for the the L-shaped domain. …”

Section: Resultsmentioning

confidence: 99%

A primal discontinuous Petrov‐Galerkin Finite Element Method for linear elasticity

Steiner

Wriggers

2017

Proc Appl Math and Mech

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Volume-locking is a significant problem when using the standard Finite Element Method (FEM) since this phenomenon can lead to failure of a numerical method. To circumvent constraints as related in (nearly) incompressible materials the novel discontinuous Petrov-Galerkin (dPG) FEM is introduced for linear elasticity. A first result of the implemented primal dPG FEM is shown for a patch test. Further tests on the performance of the proposed method are work in progress and are not part of this contribution.

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Section: Resultsmentioning

confidence: 99%

A primal discontinuous Petrov‐Galerkin Finite Element Method for linear elasticity

Steiner

Wriggers

2017

Proc Appl Math and Mech

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“…For the case of variable material coefficients, see [1]. We remark that the proposed Petrov-Galerkin shares some properties with the discontinuous Petrov-Galerkin method [2,3], and it appears worth investigating these connections in future work.…”

Section: Definition Of the Methodsmentioning

confidence: 99%

Multiscale Petrov‐Galerkin FEM for Acoustic Scattering

Gallistl

Peterseim

Carstensen

2016

Proc Appl Math and Mech

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We present a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ. We use standard continuous Q1 finite elements at a coarse discretization scale H as trial functions. The test functions are the solutions of local problems at a finer scale h. The diameter of the support of the test functions behaves like mH for some oversampling parameter m. Provided m is of the order of log(κ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ −1 .

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“…It would be interesting topic to modify the a posteriori error estimators developed in this paper for DPF method. We refer the interested readers to [12][13][14][15][16][17][18] and references therein for the DPG method and its applications to various problems.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global L2 norm

Cai¹,

Carey²,

Ku³

et al. 2015

Computers & Mathematics with Applications

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a b s t r a c tA new asymptotically exact a posteriori error estimator is developed for first-order div least-squares (LS) finite element methods. Let (u h , σ h ) be LS approximate solution for (u , σ = −A∇u). Then, E = ∥A −1/2 (σ h + A∇u h )∥ 0 is asymptotically exact a posteriori error estimator for ∥A 1/2 ∇(u − u h )∥ 0 or ∥A −1/2 (σ − σ h )∥ 0 depending on the order of approximate spaces for σ and u. For E to be asymptotically exact for ∥A 1/2 ∇(u − u h )∥ 0 , we require higher order approximation property for σ, and vice versa. When both A∇u and σ are approximated in the same order of accuracy, the estimator becomes an equivalent error estimator for both errors. The underlying mesh is only required to be shape regular, i.e., it does not require quasi-uniform mesh nor any special structure for the underlying meshes. Confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for (u h , σ h ).

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Low-order dPG-FEM for an elliptic PDE

Cited by 22 publications

References 6 publications

A primal discontinuous Petrov‐Galerkin Finite Element Method for linear elasticity

A primal discontinuous Petrov‐Galerkin Finite Element Method for linear elasticity

Multiscale Petrov‐Galerkin FEM for Acoustic Scattering

Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global L2 norm

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