The Gradient Discretisation Method

Droniou, Jérôme; Eymard, Robert; Gallouët, Thierry; Guichard, Cindy; Herbin, Raphaèle

doi:10.1007/978-3-319-79042-8

Cited by 138 publications

(278 citation statements)

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“…We consider here polygonal or polyhedral meshes corresponding to couples M h (T h , F h ), where T h is a finite collection of polygonal elements such that h max T ∈ T h h T > 0 with h T denoting the diameter of T, while F h is a finite collection of hyperplanar faces. It is assumed henceforth that the mesh M h matches the geometrical requirements detailed in [22,Definition 7.2]; see also [21,Section 2]. To avoid dealing with jumps of the permeability coefficient inside elements, we additionally assume that M h is compliant with the partition P Ω on which κ is piecewise constant meaning that, for every T ∈ T h , there exists a unique subdomain ω ∈ P Ω such that T ⊂ ω.…”

Section: Space Meshmentioning

confidence: 99%

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A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Botti

Pietro

Sochala

2019

Computational Methods in Applied Mathematics

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In this work we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

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Section: Space Meshmentioning

confidence: 99%

“…The proof of the following theorem hinges on the arguments of [13,Theorem 3.3]. where C cv is the coercivity constant of σ (see (2b)), and we denote by a lin h the bilinear form obtained by replacing σ with σ lin in (22). We consider the following auxiliary linear problem: For all 1 ≤ n ≤ N,…”

Section: Stability and Well-posednessmentioning

confidence: 99%

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Botti

Pietro

Sochala

2019

Computational Methods in Applied Mathematics

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“…In most situations, the C 2 regularity on T can be weakened to an H 2 regularity, upon additional technicalities that we do not address here to simplify the exposition. See, e.g., [13,Section 7.4] for lemmas useful for establishing consistency estimates under H 2 -regularity of the function.…”

Section: Error Estimatementioning

confidence: 99%

“…The coercivity of the HMM fluxes result from the construction of the method as a Gradient Discretisation Method, see [13, Chapter 13] -we note that this coercivity is purely algebraic, and not impacted by the curvature of the faces σ ∈ S Γ or of their edges. In the case of flat faces and edges, the consistency of the fluxes (41) is a consequence of (40), see [13,Chapter 13] or [9,Example 31].…”

Section: B1 Boundary Fluxesmentioning

confidence: 99%

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Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

Droniou

Medl’a

Mikula

2019

Journal of Computational Physics

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We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma [9], is conducted in this generic finite volume framework, and yields the expected O(h) optimal convergence rate in discrete energy norm.We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in [29]. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.

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A variational discrete element method for quasistatic and dynamic elastoplasticity

Marazzato

Ern

Monasse

2020

Numerical Meth Engineering

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Summary We propose a new discrete element method supporting general polyhedral meshes. The method can be understood as a lowest‐order discontinuous Galerkin method parametrized by the continuous mechanical parameters (Young's modulus and Poisson's ratio). We consider quasistatic and dynamic elastoplasticity, and in the latter situation, a pseudoenergy conserving time‐integration method is employed. The computational cost of the time‐stepping method is moderate since it is explicit and used with a naturally diagonal mass matrix. Numerical examples are presented to illustrate the robustness and versatility of the method for quasistatic and dynamic elastoplastic evolutions.

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The Gradient Discretisation Method

Cited by 138 publications

References 0 publications

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

A variational discrete element method for quasistatic and dynamic elastoplasticity

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