A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

Cui, Jintao; Gao, Fuzheng; Sun, Zhengjia; Zhu, Peng

doi:10.1002/num.22443

Cited by 6 publications

(4 citation statements)

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“…Polygonal/polyhedral meshes don't require any postprocessing within the adaptive procedure, so polygonal/polyhedral mesh refinement is more popular than traditional triangular/tetrahedral mesh one. Several numerical discretization methods which admit polygonal/polyhedral meshes have been proposed within the current literatures; here we mention DG-FEMs [12], Mimetic Finite Difference (MFD) methods [4], Hybrid High-Order (HHO) methods [49], locally conservative methods [36], virtual FEMs [25,38], finite volume methods [8], WG-FEMs [1,21,27], and so forth. HDG, HHO and WG-FEM are closely related, share many aspects in common and are equivalent in some special cases.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

2024

Numerical Methods Partial

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In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.

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Section: Introductionmentioning

confidence: 99%

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

2024

Numerical Methods Partial

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“…His articles focus mainly on the LDG methods for one-dimensional or two-dimensional domains discretised with Cartesian grids. Another approach is proposed in 62 where the error in the elements is based on the oscillations in the right-hand side function when projected to the approximation space.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

Jaśkowiec

Pamin

2023

Sci Rep

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The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson’s and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement.

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“…Residual-type a posteriori error bounds for interior penalty dG methods on composite/polytopic meshes appeared in [26,18]. Also, in the context of virtual element methods, corresponding bounds are proven in [12,16], while for the weak Galerkin approach, an a posteriori error analysis can be found in [34].…”

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confidence: 99%

“…A number of related results followed, improving various aspects of the theory; for instance, see [3,28,15,40,1,6,27,20,32]. A key reason for the aforementioned restrictive assumption that all elemental faces are of comparable size to the element diameter in existing a posteriori error analysis for polytopic dG methods [26,18] is exactly the lack fo availability of a stability result corresponding to [30,Theorem 2.2] for polytopic element meshes containing elements with small faces.…”

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confidence: 99%

A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes

Cangiani¹,

Dong²,

Georgoulis³

2022

Preprint

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We present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds are also applicable to dG methods on meshes consisting of elements with very general polygonal/polyhedral shapes. The case of simplicial and/or box-type elements is included in the analysis as a special case. In particular, for the upper bounds, arbitrary number of very small faces are allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptions are satisfied. As a corollary, the present analysis generalizes known a posteriori error bounds for dG methods, allowing in particular for meshes with arbitrary number of irregular hanging nodes per element. The proof hinges on a new conforming recovery strategy in conjunction with a Helmholtz decomposition formula. The resulting a posteriori error bound involves jumps on the tangential derivatives along elemental faces. Local lower bounds are also proven for a number of practical cases. Numerical experiments is also presented, highlighting the practical value of the derived a posteriori error bounds as error estimators.

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A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

Cited by 6 publications

References 50 publications

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes

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