Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type

Cancès, Clément; Droniou, Jérôme; Guichard, Cindy; Manzini, Gianmarco; Olivares, Manuela Bastidas; Pop, Sorin

doi:10.1007/978-3-030-69363-3_2

Cited by 2 publications

(2 citation statements)

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“…Error estimates could also be stated, but they are more complicated to present and require stronger assumptions on the solution to the Stefan equation; we therefore refer the interested reader to [10] for details, in which a partial uniqueness result is also stated for the solution of (15). We also mention in passing that error estimates for transient Stefan/porous medium equations are established in [4]; these estimates are stated in the generic framework of the Gradient Discretisation Method, which covers polytopal nonconforming methods.…”

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confidence: 99%

“…We note that some of the tests here involve non-homogeneous Dirichlet boundary conditions; adapting the LEPNC scheme to this case is straightforward, and done as for standard non-conforming P 1 finite elements. We also refer the interested reader to [4] for a numerical assessment of the LEPNC (and comparison with other methods) on the transient porous medium equation.…”

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confidence: 99%

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Non-conforming finite elements on polytopal meshes

Droniou,

Eymard,

Gallouet

et al. 2020

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In this work we present a generic framework for non-conforming finite elements on polytopal meshes, characterised by elements that can be generic polygons/polyhedra. We first present the functional framework on the example of a linear elliptic problem representing a single-phase flow in porous medium. This framework gathers a wide variety of possible nonconforming methods, and an error estimate is provided for this simple model. We then turn to the application of the functional framework to the case of a steady degenerate elliptic equation, for which a mass-lumping technique is required; here, this technique simply consists in using a different -piecewise constant-function reconstruction from the chosen degrees of freedom. A convergence result is stated for this degenerate model. Then, we introduce a novel specific non-conforming method, dubbed Locally Enriched Polytopal Non-Conforming (LEPNC). These basis functions comprise functions dedicated to each face of the mesh (and associated with average values on these faces), together with functions spanning the local P 1 space in each polytopal element. The analysis of the interpolation properties of these basis functions is provided, and mass-lumping techniques are presented. Numerical tests are presented to assess the efficiency and the accuracy of this method on various examples. Finally, we show that generic polytopal non-conforming methods, including the LEPNC, can be plugged into the gradient discretization method framework, which makes them amenable to all the error estimates and convergence results that were established in this framework for a variety of models.

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confidence: 99%

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confidence: 99%