On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

Bassi, Francesco; Botti, Lorenzo Alessio; Colombo, Alessandro; Pietro, Daniele Antonio Di; Tesini, Pietro

doi:10.1016/j.jcp.2011.08.018

Cited by 227 publications

(192 citation statements)

References 16 publications

(27 reference statements)

Supporting

Mentioning

191

Contrasting

Order By: Relevance

“…Following the approach presented in [4], for each equation of the system, and without loss of generality, we choose the set of test and shape functions in any element K coincident with the set {φ} of N K dof orthogonal and hierarchical basis functions in that element. Such basis is built by means of the modified Gram-Schmidt (MGS) algorithm, starting from a set of monomials defined over each elementary space P k d (K) in a reference frame relocated in the element barycenter and aligned with the principal axes of inertia of K.…”

Section: Dg Space Discretizationmentioning

confidence: 99%

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Massa¹,

Noventa²,

Bassi³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

View full text Add to dashboard Cite

Abstract. In recent years the increasing attention to high-order Finite Volume (FV), Finite Element (FE) and spectral methods and the growth of computing power promote the development of high-order temporal schemes to perform robust, accurate and efficient unsteady long-time simulations. In this context, some features of the Discontinuous Galerkin finite element (DG) methods, e.g. compactness and flexibility, can be advantageous both for explicit and implicit time integration approaches. Explicit schemes can achieve very high accuracy, but are limited by time-step restrictions, while implicit schemes, even if memory consuming, can overcome time-step limitations, thus improving the time integration efficiency. During last decades several high order implicit temporal schemes have been developed, and some of them have been successfully coupled with DG methods. However these schemes can show the order reduction if applied to very stiff problems or problems with time-dependent boundary conditions. To overcome these limitations, high-order linearly implicit two-step peer methods have been proposed and successfully applied to the numerical solution of differential-algebraic equations. The aim of the present work is to implement high-order two-step peer methods in a DG code and assess their performance for the unsteady solution of the incompressible Navier-Stokes (INS) and Reynolds Averaged Navier-Stokes equations (RANS) equations.

show abstract

Section: Dg Space Discretizationmentioning

confidence: 99%

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Massa¹,

Noventa²,

Bassi³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

View full text Add to dashboard Cite

show abstract

Section: To Cite This Versionmentioning

confidence: 99%

“…in sedimentary basin modeling non-standard elements and nonconformities can appear due to the erosion of geological layers; -(ii) the number of degrees of freedom can be reduced by aggregative coarsening techniques -cf. [7] for an application in the context of discontinuous Galerkin (dG) methods; -(iii) geometrical features can be represented more accurately without unduly increasing the number of mesh elements.Handling general polyhedral meshes requires numerical schemes that possess the usual properties of stability and consistency. In the context of cell centered finite volume methods, a popular way to achieve consistency on general polyhedral meshes is provided by Multipoint Finite Volume schemes independently introduced by Aavatsmark, Barkve, Bøe and Mannseth [2] and Edwards and Rogers [22].…”

mentioning

confidence: 99%

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

2012

View full text Add to dashboard Cite

Abstract. In this work we propose an original implementation of a large family of lowest-order methods for diffusive problems including standard and hybrid finite volume methods, mimetic finite difference-type schemes, and cell centered Galerkin methods. The key idea is to regard the method at hand as a (Petrov-)Galerkin scheme based on possibly incomplete, broken affine spaces defined from a gradient reconstruction and a point value. The resulting unified framework serves as a basis for the development of a FreeFEM-like domain specific language targeted at defining discrete linear and bilinear forms. Both the back-end and the front-end of the language are extensively discussed, and several examples of applications are provided. The overhead of the language is evaluated by comparing with a more traditional implementation. A benchmark including the comparison with more classical finite element methods on standard meshes is also proposed.Key words. Domain specific embedded language, finite volume methods, cell centered Galerkin methods, Petrov-Galerkin methods AMS subject classifications. 65N08, 65N30, 65Y051. Introduction. Lowest-order methods possibly featuring conservation of physical quantities are traditionally employed in industrial applications where computational cost is a crucial issue. In this context, the use of general polyhedral, possibly nonconforming meshes commends itself for a number of reasons. To cite a few: (i) remeshing can be avoided or postponed in problems that involve mesh deformation -e.g. in sedimentary basin modeling non-standard elements and nonconformities can appear due to the erosion of geological layers; -(ii) the number of degrees of freedom can be reduced by aggregative coarsening techniques -cf. [7] for an application in the context of discontinuous Galerkin (dG) methods; -(iii) geometrical features can be represented more accurately without unduly increasing the number of mesh elements.Handling general polyhedral meshes requires numerical schemes that possess the usual properties of stability and consistency. In the context of cell centered finite volume methods, a popular way to achieve consistency on general polyhedral meshes is provided by Multipoint Finite Volume schemes independently introduced by Aavatsmark, Barkve, Bøe and Mannseth [2] and Edwards and Rogers [22]. The main advantage of multipoint schemes is that they can be easily fitted into existing simulators based on standard finite volume schemes. A major drawback is their lack of stability in some configurations. Two ways of overcoming this difficulty by designing discretizations based on the variational formulation of the problem and featuring cell-and face-unknowns have been proposed by Brezzi, Lipnikov and coworkers [8,9] (Mimetic Finite Difference methods) and by Droniou and Eymard [20] (Mixed/Hybrid Finite Volume methods). In this context, Eymard, Gallouët and Herbin [24] have shown that face unknowns can be selectively used as Lagrange multipliers to enforce flux continuity, or eliminated using a consisten...

show abstract

“…In the context of discretizing PDEs in complicated geometries, Composite Finite Elements (CFEs) have been developed in the articles [33,32] and [1,31] for both conforming finite element and discontinuous Galerkin (DGFEM) methods, respectively, which exploit general meshes consisting of agglomerated elements consisting of a collection of neighbouring elements present within a standard finite element method. A closely related technique based on employing the so-called agglomerated DGFEM has also been considered in [7,8,9]. From a meshing point of view, the exploitation of general polytopic elements provides enormous flexibility.…”

Section: Introductionmentioning

confidence: 99%

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

118

View full text Add to dashboard Cite

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.

show abstract

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

Cited by 227 publications

References 16 publications

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

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

Contact Info

Product

Resources

About