SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Analysis of Compatible Discrete Operator Schemes for Stokes problem on Polyhedral Meshes

J. Bonelle and A. Ern

Compatible Discrete Operator (CDO) schemes belong to the class of compatible (or mimetic, or structure-preserving) schemes. Their aim is to preserve key structural properties of the underlying PDE. This is achieved by distinguishing topological laws and constitutive relations. CDO schemes are formulated using discrete differential operators for the topological laws and discrete Hodge operators for the constitutive relations. CDO schemes have been recently analyzed in [1] for elliptic problems. We first review the main results in this case. Design properties for discrete Hodge operators leading to stability and $P_0$-consistency are identified. We also highlight links between CDO schemes and existing schemes in the litterature [2, 3]. Then, we derive CDO schemes for Stokes flows that are closely related to the recent work of Kreeft and Gerritsma [4]. Finally, we present numerical results on 3D test cases.

[1] J. Bonelle and A. Ern, Analysis of Compatible Discrete Operator schemes for Elliptic Problems on Polyhedral Meshes, Submitted. Available from: (2012).
[2] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite difference for elliptic problem, M2AN, 43 (2009).
[3] R. Eymard, C. Guichard and R. Herbin, Small stencil 3D schemes for diffusive flows in porous media, M2AN, 46 (2012).
[4] J. Kreeft and M. Gerritsma, A priori error estimates for compatible spectral discretization of the Stokes problem for all admissible boundary conditions, arxiv:1206.2812 (2012).

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