Recent techniques for PDE discretizations on polyhedral meshes

Bellomo, Nicola; Brezzi, Franco; Manzini, Gianmarco

doi:10.1142/s0218202514030018

Cited by 18 publications

(14 citation statements)

References 18 publications

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“…Construction of the load term. Let Π K denote the L 2 -projection onto P (K) and f h be the piecewise polynomial approximation of f on T h given by (16) f…”

Section: 4mentioning

confidence: 99%

The fully nonconforming virtual element method for biharmonic problems

Antonietti

Manzini

Verani

2017

Math. Models Methods Appl. Sci.

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146

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In this paper we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

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“…Construction of the load term. Let Π K denote the L 2 -projection onto P (K) and f h be the piecewise polynomial approximation of f on T h given by (16) f…”

Section: 4mentioning

confidence: 99%

The fully nonconforming virtual element method for biharmonic problems

Antonietti

Manzini

Verani

2017

Math. Models Methods Appl. Sci.

Self Cite

146

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“…7.1.4. The state of the art on these topics is well represented in two recent special issues on numerical methods for PDEs on unstructured meshes [42,41].…”

Section: 5mentioning

confidence: 99%

Annotations on the virtual element method for second-order elliptic problems

Manzini

2017

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stiffness matrix is rank deficient). In the VEM we fix this issue by adding a stabilization term to the bilinear form. Stabilization terms are required not to upset the exactness property for polynomials and to scale properly with respect to the mesh size and the problem coefficients.1.4.1. The low-order setting for polygonal cells (in 2D) and polyhedral cells (in 3D) can be obtained as a straightforward generalization of the FEM. The high-order setting deserves special care in the treatment of variable coefficients in order not to loose the optimality of the discretization and in the 3D formulation.1.5. The shape functions are virtual in the sense that we never compute them explicitly (not even approximately) inside the elements. Note that the polynomial projection of the approximate solution is readily available from the degrees of freedom and we can use it as the numerical solution.1.6. The major advantage offered by the VEM is in the great flexibility in admitting unstructured polygonal and polyhedral meshes.1.7. The VEM can be seen as an evolution of the MFD method. Background material (and a few historical notes) will be given in the final section of this document.

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“…Our aim is to use it together with the finite volume scheme on the same spatial grid. Several such schemes have been identified in recent reviews [8]. Without being exhaustive, we may mention the extension of finite element methods to polyhedral meshes with the use of barycentric [27] or harmonic [11] shape func-tions, the families of Discontinuous Galerkin [24], Hybrid Discontinuous Galerkin [15] and Hybrid High Order [16] methods, the Weak Galerkin method [29] or the Mimetic Finite Differences (MDF) [28].…”

Section: Introductionmentioning

confidence: 99%

A fully coupled scheme using virtual element method and finite volume for poroelasticity

et al. 2019

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In this paper, we design and study a fully coupled numerical scheme for the poroelasticity problem modelled through Biot's equations. The classical way to numerically solve this system is to use a finite element method for the mechanical equilibrium equation and a finite volume method for the fluid mass conservation equation. However, to capture specific properties of underground media such as heterogeneities, discontinuities and faults, meshing procedures commonly lead to badly shaped cells for finite element based modelling. Consequently, we investigate the use of the recent virtual element method which appears as a potential discretization method for the mechanical part and could therefore allow the use of a unique mesh for the both mechanical and fluid flow modelling. Starting from a first insight into virtual element method applied to the elastic problem in the context of geomechanical simulations, we apply in addition a finite volume method to take care of the fluid conservation equation. We focus on the first order virtual element method and the two point flux approximation for the finite volume part. A mathematical analysis of this original coupled scheme is provided, including existence and uniqueness results and a priori estimates. The method is then illustrated by some computations on two or three dimensional grids inspired by realistic application cases.

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Recent techniques for PDE discretizations on polyhedral meshes

Cited by 18 publications

References 18 publications

The fully nonconforming virtual element method for biharmonic problems

The fully nonconforming virtual element method for biharmonic problems

Annotations on the virtual element method for second-order elliptic problems

A fully coupled scheme using virtual element method and finite volume for poroelasticity

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