Local flux mimetic finite difference methods

Lipnikov, Konstantin; Shashkov, Mikhail; Yotov, Ivan

doi:10.1007/s00211-008-0203-5

Cited by 117 publications

(88 citation statements)

References 42 publications

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“…In this paper we introduce a finite volume methodology for simulation of diffusion process on general evolving polygonal meshes. our finite volume is closely related to the pioneer work by C. Le Potier in [3] and the work of K. Lipnikov, M. Shashkov and I. Yotov in [4].…”

Section: Introductionmentioning

confidence: 71%

A Convergent Finite Volume Type O-method on Evolving Surfaces

Nemadjieu

2010

AIP Conference Proceedings

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Abstract. We present a finite volume scheme for anisotropic diffusion on evolving hypersurfaces. The underlying motion is assumed to be described by a fixed, not necessarily normal, velocity field. The ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating polygonal meshes, where grid vertices move on motion trajectories, a consistent finite volume discretization of the induced transport on the cells (polygonal patches), and a proper incorporation of a diffusive flux balance at polygonal faces. The main stability results and convergence estimate are obtained.

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Section: Introductionmentioning

confidence: 71%

A Convergent Finite Volume Type O-method on Evolving Surfaces

Nemadjieu

2010

AIP Conference Proceedings

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“…The stress continuity condition (16) implies that t K,K ,s,σ whenever σ = K ∩ K . The system of equations for linear elasticity are then given by the discrete conservation of momentum (13), the definition of the force on faces (12) and the multi-point approximation of sub-face forces given by (18). We have represented the points that are used to define the discrete gradient with a green contour.…”

Section: 2mentioning

confidence: 99%

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

2017

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Abstract. In this paper we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regards to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular we look how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.

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“…This was first done by [21,24] and is called the local-flux mimetic formulation of the MPFA method. In this case, all faces of a cell are such that each corner is associated with d unique faces, where d is the dimension.…”

Section: Raviart-thomas (Rt0)mentioning

confidence: 99%

Open-source MATLAB implementation of consistent discretisations on complex grids

et al. 2011

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Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains addon modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

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Local flux mimetic finite difference methods

Cited by 117 publications

References 42 publications

A Convergent Finite Volume Type O-method on Evolving Surfaces

A Convergent Finite Volume Type O-method on Evolving Surfaces

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

Open-source MATLAB implementation of consistent discretisations on complex grids

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