Error Estimates for a Class of Degenerate Parabolic Equations

Ebmeyer, Carsten

doi:10.1137/s0036142996305200

Cited by 32 publications

(31 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We assume in the next that the fully discrete schemes above have a unique solution and we refer to [2,3,9,24] for a proof.…”

Section: Linearization Methods For Richards' Equationmentioning

confidence: 99%

“…As regards the spatial discretization there are much more options possible. Galerkin finite elements were used in [2,3,9,22,24,32], often together with mass lumping to ensure a maximum principle [8]. Locally mass conservative schemes for Richards' equation were proposed and analysed in [10,11] (finite volumes), in [16] (multipoint flux approximation) or [4,5,26,29,34,35] (mixed finite element method).…”

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

confidence: 99%

See 1 more Smart Citation

A study on iterative methods for solving Richards’ equation

2016

View full text Add to dashboard Cite

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.

show abstract

“…We assume in the next that the fully discrete schemes above have a unique solution and we refer to [2,3,9,24] for a proof.…”

Section: Linearization Methods For Richards' Equationmentioning

confidence: 99%

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

confidence: 99%

A study on iterative methods for solving Richards’ equation

2016

View full text Add to dashboard Cite

show abstract

“…For the numerical analysis of conformal discretizations of the Richards equation in the pressure formulation we refer to [16,21] and in the saturation based formulation to [23], where both type of degeneracy are allowed but the results do not apply to the fully saturated flow regime. We mention [7] for the equilibrium sorption transport problem, as well as [14,24] for the porous medium equation.…”

Section: Remark 13mentioning

confidence: 99%

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

2008

View full text Add to dashboard Cite

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one ) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart-Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.

show abstract

“…Due to the special structure of (3.6), the usual conditions on the acuteness of the triangulation (see [11]) are not needed in order to prove this semi-discrete maximum principle.…”

Section: The Regularized Problemmentioning

confidence: 99%

The smoothing property for a class of doubly nonlinear parabolic equations

Ebmeyer

Urbano

2005

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well-known feature in some particular cases such as the porous medium equation or the parabolic p-Laplace equation. The result is obtained via regularization and a comparison theorem.

show abstract

Error Estimates for a Class of Degenerate Parabolic Equations

Cited by 32 publications

References 9 publications

A study on iterative methods for solving Richards’ equation

A study on iterative methods for solving Richards’ equation

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

The smoothing property for a class of doubly nonlinear parabolic equations

Contact Info

Product

Resources

About