Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces

By Robert Eymard, T. Gallouet, Raphaèle Herbin. 2008

392 citations recorded by the source corpus.

A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/63b73ec0c8de2ea8c371790509b852ebe473ab8a

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In Computational Mechanics

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Citations in the corpus

Vertex-centred discretization of multiphase compositional Darcy flows on general meshes (2012, methodology): “Among these, let us mention for example the well-known O scheme introduced in [1], [2], [17], [18], the L scheme [3], or the Sushi scheme [19].”
A monotone nonlinear finite volume method for approximating diffusion operators on general meshes (2016, methodology): “Nowadays, besides the numerical methods that are currently used for the approximation of isotropic or anisotropic diffusion operators on rectangle, triangle (tetrahedron), or quadrilateral (hexahedron) meshes — finite difference (FD), see [1], — conformal and non-conformal finite element (FE), see [”
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods (2016, methodology): “This case is closely related (up to an equivalent choice of the stabilization) to Hybrid Finite Volumes [22].”
Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model (2012, methodology): “The estimate for k is then obtained by testing (31) against a nonlinear function of k lying in L(Q); this (rather technical) proof involves in particular discrete Sobolev inequalities [6, 7], and may be found in [9].”
Small-stencil 3D schemes for diffusive flows in porous media (2012, methodology): “Remark 1 (Error estimate) In the case where u and Λ are sufficiently regular, we get for the SUSHI scheme [18] and for the schemes presented in this paper, that WD(Λ∇u) ≤ ChD and SD(u) ≤ ChD, for C > 0 depending on the regularity of the mesh.”
MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS (2013, methodology): “The last goal of the present paper is to recall known and show new relations between MFE methods and other discretization schemes, namely the two-point finite volume (FV) [29], mimetic finite difference (MFD) [17, 13], hybrid finite volume [31], mixed finite volume [26], multi-point flux-approximati”