Mixed finite elements for the Richards’ equation: linearization procedure

Pop, Sorin; Radu, Florin A.; Knabner, Peter

doi:10.1016/j.cam.2003.04.008

Cited by 154 publications

(154 citation statements)

References 14 publications

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“…The method still involves the computation of derivatives and in the degenerate case might also fail to converge (see the numerical examples in Section 4). The L−method was proposed for Richards' equation by [25,32,36] and it is the only method which exploits the monotonicity of θ(·). The L−scheme to solve the non-linear problem (4) reads:…”

Section: Linearization Methods For Richards' Equationmentioning

confidence: 99%

“…Moreover, the schemes [25,36] are considering the Kirchhoff transformation, which is not the case in the present work.…”

Section: Remarkmentioning

confidence: 99%

“…It was proposed for Richards' equation in [25,32,36]. The method is robust and linearly convergent, and does not involve the computation of any derivatives.…”

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

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A study on iterative methods for solving Richards’ equation

2016

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This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards' equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards' equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L−scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L−scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L−scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.

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Section: Linearization Methods For Richards' Equationmentioning

confidence: 99%

“…Moreover, the schemes [25,36] are considering the Kirchhoff transformation, which is not the case in the present work.…”

Section: Remarkmentioning

confidence: 99%

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A study on iterative methods for solving Richards’ equation

2016

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“…Therefore to solve (5.2) we use a linear iteration scheme inspired by the L-scheme discussed in [36,38]. Specifically, for a sufficiently large L that will be specified later and with i as the iteration index, we solve the linear elliptic equation…”

Section: Numerical Schemementioning

confidence: 99%

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Duijn

Mitra

Pop

2018

Nonlinear Analysis: Real World Applications

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The Richards equation is a mathematical model for the unsaturated flow through porous media. This paper considers an extension of the Richards equation, where non-equilibrium effects like hysteresis and dynamic capillarity are incorporated in the relationship that relates the water pressure and the saturation. The focus is on travelling wave solutions, for which the existence is investigated first for the model including hysteresis and subsequently for model including dynamic capillarity effect. In particular, such solutions may have non monotonic profiles, which are ruled out when considering standard, equilibrium type models, but have been observed experimentally. The paper ends with numerical experiments confirming the theoretical results. In this sense the original system of partial differential equations is solved by means of an implicit scheme. This relies on an equivalent, mixed formulation of the system. For solving the resulting nonlinear, time-discrete problems, a linear iterative scheme is proposed.

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“…In [20], this approach is placed in a fixed point context, for approximating the solution of an elliptic problem with a nonlinear and possibly unbounded source term (see also [27]). The same ideas are followed in [23] and [19], where similar schemes are considered for the implicit discretization of a degenerate (fast diffusion) problem in both conformal and mixed formulation. We also mention here [24] for a related work on nonlinear elliptic equations.…”

Section: A Fixed Point Iteration For the Time Discrete Problemsmentioning

confidence: 99%

A Numerical Scheme for the Pore-Scale Simulation of Crystal Dissolution and Precipitation in Porous Media

Devigne¹,

Pop²,

Duijn³

et al. 2008

SIAM J. Numer. Anal.

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Abstract.In this paper we analyze a numerical scheme for a dissolution and precipitation in porous media. We focus here on the chemistry, which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary. We use a regularization approach for constructing a semi-implicit scheme that is stable and convergent. For dealing with the emerging time discrete nonlinear problems, we propose a simple fixed point iterative procedure. The paper is concluded by numerical results.

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Mixed finite elements for the Richards’ equation: linearization procedure

Cited by 154 publications

References 14 publications

A study on iterative methods for solving Richards’ equation

A study on iterative methods for solving Richards’ equation

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

A Numerical Scheme for the Pore-Scale Simulation of Crystal Dissolution and Precipitation in Porous Media

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