Séance Séminaire

Séminaire ACSIOM

High order mimetic discretization

In this work the High Order Mimetic Discretization Framework will be presented together with a discussion of two fundamental aspects for the construction of structure-preserving discretizations: (i) the definition of the discrete degrees of freedom of physical field quantities, and (ii) the formulation of the physical field laws. For the first, the geometric degrees of freedom will be introduced. These degrees of freedom are associated to geometric objects (points, lines, surfaces and volumes), and a relation to differential forms will be remarked. It will be shown that it is possible to construct discrete polynomial function spaces of arbitrary degree associated to these geometric degrees of freedom. Moreover, these function spaces constitute a discrete de Rham complex. In this way it is possible to exactly discretize topological equations even on highly deformed meshes. All approximation errors are included in the constitutive equations. This leads to discretizations that exactly preserve the divergence free constraint of velocity fields in incompressible flow problems and of magnetic fields in electromagnetic problems, for example. For the second, the Navier-Stokes equations will be used as an example and we will show that although at the continuous level all equivalent formulations are equally good, at the discrete level, the choice of a particular formulation has a fundamental impact on the conservation properties of the discretization. These ideas will be illustrated with the application to the solution of: e.g. Poisson equation [1], Darcy flow, fusion plasma equilibrium [2], and Navier-Stokes equations [3]. References [1] A. Palha, P. P. Rebelo, R. Hiemstra, J. Kreeft, and M. Gerritsma. Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms?. In: Journal of Computational Physics 257 (2014), pp. 1394?1422. DOI: 10.1016/j.jcp.2013.08.005. [2] A. Palha, B. Koren, and F. Felici. A mimetic spectral element solver for the Grad?Shafranov equation?. In: Journal of Computational Physics 316 (2016), pp. 63?93. DOI: 10.1016/j.jcp.2016.04.002. [3] A. Palha and M. Gerritsma. A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations?. In: Journal of Computational Physics 328 (2017), pp. 200?220. DOI: 10.1016/j.jcp.2016.10.009.