<i>hp</i>-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes

Cangiani, Andrea; Dong, Zhaonan; Georgoulis, Emmanuil H.; Houston, Paul

doi:10.1051/m2an/2015059

Cited by 60 publications

(141 citation statements)

References 45 publications

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“…Following [20,21], we assume that a sub-triangulation into faces of each mesh interface is given if d = CFE is the union of element boundary faces, i.e., F ⊂ ∂ Ω for F ∈ F B CFE . The boundary ∂ κ of an element κ and the sets ∂ κ \ ∂ Ω and ∂ κ ∩ ∂ Ω will be identified in a natural way with the corresponding subsets of F CFE .…”

Section: Discontinuous Galerkin Methods On Polytopic Meshesmentioning

confidence: 99%

“…Following [20], for the case when general polytopic elements are admitted, in the absence of optimal hp-approximation results for the local L 2 -projection operator, we prove an inf-sup condition for the bilinear formB Hyp (·, ·), with respect to the following streamline DGFEM-norm:…”

Section: Error Analysismentioning

confidence: 99%

“…One of the most popular software packages employed for this purpose is METIS [37], cf. [20,29]. From a theoretical point of view, this setting is more difficult to analyse; we shall return to this issue in Section 3.…”

Section: With This Construction Noting That the Meshes {Mmentioning

confidence: 99%

“…We finish this section by recalling the inverse estimate for the H 1 -(semi)norm derived in [20], cf. also [3].…”

Section: Inverse Estimatesmentioning

confidence: 99%

“…With this in mind, we follow the work presented in [33,32,1]; the inclusion of general polytopic meshes which admit arbitrarily small/degenerate (d − k)-dimensional element facets, k = 1,.. ., d − 1, where d denotes the spatial dimension, will also be discussed, following [21,20]. The structure of this article is as follows.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

Antonietti

Cangiani

Collis

et al. 2016

Lecture Notes in Computational Science and Engineering

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118

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The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.

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Section: Discontinuous Galerkin Methods On Polytopic Meshesmentioning

confidence: 99%

Section: Error Analysismentioning

confidence: 99%

Section: With This Construction Noting That the Meshes {Mmentioning

confidence: 99%

“…We finish this section by recalling the inverse estimate for the H 1 -(semi)norm derived in [20], cf. also [3].…”

Section: Inverse Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

Antonietti

Cangiani

Collis

et al. 2016

Lecture Notes in Computational Science and Engineering

Self Cite

118

View full text Add to dashboard Cite

show abstract

Error estimates for completely discrete FEM in energy‐type and weaker norms

Angermann,

Knabner,

Rupp

2024

Numerical Methods Partial

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The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.

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High–order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations

Antonietti

Facciolà

Houston

et al. 2021

Polyhedral Methods in Geosciences

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We present a comprehensive review of the current development of PolyDG methods for geophysical applications, addressing as paradigmatic applications the numerical modeling of seismic wave propagation and fracture reservoir simulations. We first recall the theoretical background of the analysis of PolyDG methods and discuss the issue of its efficient implementation on polytopic meshes. We address in detail the issue of numerical quadrature and recall the new quadrature free algorithm for the numerical evaluation of the integrals required to assemble the mass and stiffness matrices introduced in [22]. Then we present PolyDG methods for the approximate solution of the elastodynamics equations on computational meshes consisting of polytopic elements. We review the well-posedness of the numerical formulation and hp-version a priori stability and error estimates for the semi-discrete scheme, following [10]. The computational performance of the fully-discrete approximation obtained based on employing the PolyDG method for the space discretization coupled with the leap-frog time marching scheme are demonstrated through numerical experiments. Next, we address the problem of modeling the flow in a fractured porous medium and we review the unified construction and analysis of PolyDG methods following [16]. We show, in a unified setting, the well-posedness of the numerical formulations and hp-version a priori error bounds, that are then validated through numerical tests. We also briefly discuss the extendability of our approach to handle networks of partially immersed fractures and networks of intersecting fractures, recently proposed in [15].

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hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes

Cited by 60 publications

References 45 publications

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

Error estimates for completely discrete FEM in energy‐type and weaker norms

High–order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations

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