A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients

Harder, Christopher; Paredes, Diego; Valentin, Frédéric

doi:10.1016/j.jcp.2013.03.019

Cited by 80 publications

(84 citation statements)

References 29 publications

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“…Parallel computed can be easily adopted for the local problems. We remark that the MMMFEM shares some similarities with the Multiscale Hybrid-Mixed finite element methods [4,20], both methods involve solving some local problems and a small dimension global problem. However, the local problems in the Multiscale Hybrid-Mixed finite element methods are Neumann boundary condition problems.…”

Section: Interface Problemmentioning

confidence: 96%

A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media

Chung

2019

Journal of Computational Physics

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In this paper, we propose a local-global multiscale mortar mixed finite element method (MMM-FEM) for multiphase transport in heterogeneous media. We consider the two-phase flow system, the pressure equation is solved via the multiscale mortar mixed finite element method, a mass conservation velocity field can be obtained, then we use explicit finite volume method to solve the saturation equation. We use polynomials and multiscale basis to form the coarse mortar space. The multiscale basis is the restriction of global pressure field obtained at previous time step on the coarse interface. We solve the pressure equation on the fine grid to initialize the simulation. Numerical experiments on some benchmark 2D and 3D heterogeneous models are provided to validate the performance of our method. * Corresponding Author

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Section: Interface Problemmentioning

confidence: 96%

A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media

Chung

2019

Journal of Computational Physics

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“…Problem (55) is well-posed, since, according to (49) and (48),Ĥ T pt T , t T q is uniformly equivalent to }t T } 2 T and the inf-sup condition holds. The global (saddle point) problem resulting from the local elimination (55) has the same size and structure as that obtained with the Multiscale Hybrid-Mixed (MHM) method derived in [3,40] on simplicial meshes.…”

Section: Static Condensationmentioning

confidence: 94%

A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

Pietro

Ern

Lemaire

2016

Lecture Notes in Computational Science and Engineering

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March 10, 2016Abstract Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells (hence, the term hybrid), and these unknowns are polynomials of arbitrary order k ě 0 (hence, the term high-order). HHO methods are devised from local reconstruction operators and a local stabilization term. The discrete problem is assembled cellwise, and cell-based unknowns can be eliminated locally by static condensation. HHO methods support general meshes, are locally conservative, and allow for a robust treatment of physical parameters in various situations, e.g., heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, and advectiondominated transport. This paper reviews HHO methods for a variable-diffusion model problem with nonhomogeneous, mixed Dirichlet-Neumann boundary conditions, including both primal and mixed formulations. Links with other discretization methods from the literature are discussed.

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“…It was first introduced for the transport equation in [1,5,6,7]. The MHM method has been also applied to the elasticity problem in [4].…”

Section: Introductionmentioning

confidence: 99%

A Locking-Free MHM Method for Elasticity

Pereira¹,

Valentin²

2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Abstract. This work presents a multiscale hybrid-mixed finite element (MHM) method for the two-and three-dimensional linear elasticity problem that deals with nearly incompressible and heterogeneous isotropic materials. The starting point is a dual-hybrid form of the elasticity model defined on a coarse mesh, which is equivalent to a set of element-wise elasticity problems brought together by a face-based global formulation. Importantly, the local problems are independent to one another and determine the basis functions. Thereby, the basis naturally incorporate multiscale features of the media. This new variant of the MHM method turns out to be robust in the incompressible limit case as a result of the use of a stabilized finite element method to approximate basis functions. Some preliminary theoretical results are addressed.

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A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients

Cited by 80 publications

References 29 publications

A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media

A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media

A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

A Locking-Free MHM Method for Elasticity

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