10:40
MS21: Structure-preserving discretization methods I: Discretization methods based on exterior calculus (Part 1)
Chair: Marc Gerritsma
10:40
25 mins
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A hybrid mimetic spectral element method for the vorticity-velocity-pressure formulation of the Stokes equations
Yi Zhang, Varun Jain, Marc Gerritsma
Abstract: A mimetic spectral element method that preserves the exterior derivative operator has been developed for the vorticity-velocity-pressure (VVP) formulation of the Stokes equations [Kreeft & Gerritsma, Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution, 2013]. Its main disadvantage is the high computational cost because of the large number of degrees of freedom which means a big global system has to be solved. A way to overcome this disadvantage is to use a hybrid method, a domain decomposition method, with which the domain is first decomposed into discontinuous sub-domains and then connected by Lagrange multipliers. To solve the problem, the Lagrange multipliers can be computed firstly. The remaining local problems then become trivial. In addition, dual basis functions will be used to retrieve a higher sparsity such that the computational cost can be further decreased.
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11:05
25 mins
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The EMAC scheme and conservation laws of Navier-Stokes Galerkin discretizations
Leo Rebholz
Abstract: We study conservation properties of Galerkin methods for the incompressible Navier–Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier–Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation. Additionally, we discuss efficient discretizations and improvements in convergence analysis offered by EMAC.
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11:30
25 mins
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Accurate numerical eigenstates of the Gross-Pitaevskii equation
Bo Gervang, Christian Bach
Abstract: Please see the attached file.
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