TP or not TP, that is the question

Eymard, Robert; Gallouët, Thierry; Guichard, Cindy; Herbin, Raphaèle; Masson, Roland

doi:10.1007/s10596-013-9392-9

Cited by 44 publications

(50 citation statements)

References 16 publications

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“…First, we describe in §2.1 how the domains Ω (and then Q) has to be meshed. In particular, the mesh has to fulfill the so-called orthogonality condition so that the diffusion fluxes can be discretized using a simple two-point flux approximation [22].…”

Section: This Equation Turns Tomentioning

confidence: 99%

Improving Newton's Method Performance by Parametrization: The Case of the Richards Equation

Brenner¹,

Cancès²

2017

SIAM J. Numer. Anal.

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Abstract. The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to Richards equation a strategy that consists in defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of Richards equation is then discretized thanks to a monotone Finite Volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate non-degeneracy conditions on the parametrization that Newtons method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach.

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Section: This Equation Turns Tomentioning

confidence: 99%

Improving Newton's Method Performance by Parametrization: The Case of the Richards Equation

Brenner¹,

Cancès²

2017

SIAM J. Numer. Anal.

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“…81 To date, it is not known how to design a method that can be written as (7.1) for any mesh and tensor (as separately noticed in Ref. 70), or how to adapt the aforementioned a priori estimate techniques to schemes not having this structure . .…”

Section: Mmp Schemes By Nonlinear Corrections Of Linear Schemesmentioning

confidence: 99%

Finite volume schemes for diffusion equations: Introduction to and review of modern methods

Droniou

2014

Math. Models Methods Appl. Sci.

257

183

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We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum-maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum-maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions.

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“…. Following [44], it can be proved that this method is closely related to the so-called two-point flux approximation (TPFA) method [19].…”

Section: Mixed Finite Element Approximationmentioning

confidence: 99%

Mixed-Dimensional Geometric Multigrid Methods for Single-Phase Flow in Fractured Porous Media

Arrarás¹,

Gaspar²,

Portero³

et al. 2019

SIAM J. Sci. Comput.

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This paper deals with the efficient numerical solution of single-phase flow problems in fractured porous media. A monolithic multigrid method is proposed for solving twodimensional arbitrary fracture networks with vertical and/or horizontal possibly intersecting fractures. The key point is to combine two-dimensional multigrid components (smoother and inter-grid transfer operators) in the porous matrix with their one-dimensional counterparts within the fractures, giving rise to a mixed-dimensional multigrid method. This combination seems to be optimal since it provides an algorithm whose convergence matches the multigrid convergence factor for solving the Darcy problem. Several numerical experiments are presented to demonstrate the robustness of the monolithic mixeddimensional multigrid method with respect to the permeability of the fractures, the grid size and the number of fractures in the network. * Francisco J.

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TP or not TP, that is the question

Cited by 44 publications

References 16 publications

Improving Newton's Method Performance by Parametrization: The Case of the Richards Equation

Improving Newton's Method Performance by Parametrization: The Case of the Richards Equation

Finite volume schemes for diffusion equations: Introduction to and review of modern methods

Mixed-Dimensional Geometric Multigrid Methods for Single-Phase Flow in Fractured Porous Media

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