Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

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Session Overview
Session
MS 22b: high-order face-based discretization methods
Time:
Tuesday, 13/July/2021:
12:00pm - 2:00pm

Session Chair: Théophile Chaumont-Frelet
Session Chair: Alexandre Ern
Session Chair: Simon Lemaire
Virtual location: Zoom 5


Session Abstract

This minisymposium focuses on face-based discretization methods, i.e. Galerkin methods

with unknowns attached to the mesh faces. These methods present several advantages,

including (i) their built-in static condensation properties leading to improved performances,

(ii) their native possibility to handle high polynomial degrees of approximation, (iii) their

natural ability to preserve at the discrete level key physical features of the continuous

model (local balances, limit behaviors), as well as (iv) their capability to handle general

polytopal meshes (hence simplifying adaptive refinement/coarsening).

Examples of face-based discretization methods encompass (but are not limited to) the Hy-

bridizable Discontinuous Galerkin (HDG) method, the Hybrid High-Order (HHO) method,

or the Multiscale Hybrid-Mixed (MHM) method.

The objective of this minisymposium is twofold. On the one hand, state-of-the-art advances

in face-based discretization methods will be presented, with applications to various areas

of mechanics or physics. On the other hand, this minisymposium aims at highlighting the

connections and at building bridges between existing face-based discretization methods

and the associated communities.


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Presentations
12:00pm - 12:30pm

A face-based eight-order scheme for convection-diffusion problems and polyhedral unstructured grids

Duarte M. S. Albuquerque, Filipe J. M. Diogo

Universidade de Lisboa, Portugal

In the present work, a previous developed high-order scheme for unstructured grids is extended to convection-diffusion problems. This method is based on the weighted least-squares and reconstructs a high-order polynomial at each face centre. A scaling technique is introduced when computing the least-squares pseudo inverse matrix, resulting in lower local matrix condition number and faster inversion time. The proposed algorithm is developed and studied for second, fourth, sixth and eighth orders schemes, showing a correct convergence order, for several analytical solutions and unstructured grids.

A different stencil expansion algorithm is tested, based on face neighbours, resulting in an accuracy improvement, while decreasing the required memory for the matrix. A new weight function is studied, having as main criteria the global and local matrix condition number and the convergence order of the mean error. An analysis on how this method works with a null source term solution and perturbed grids, shows the robustness of the technique and its possible limits, regarding the convection-diffusion ratio. Finally, a study on different meshes types (Cartesian, triangular, polyhedral, hybrid and irregular quadrilaterals) and several orders is done to compare the best and most efficient combinations for regular and irregular grids.

Acknowledgement. The present work is a contribution to the research project High-order immersed boundary for moving body problems - HIBforMBP - with the reference PTDC/EME-EME/32315/2017, which is supported by FCT.



12:30pm - 1:00pm

Hybrid High-Order methods for creeping flows of non-Newtonian fluids

Michele Botti1, Daniel Castanon Quiroz2, Daniele A. Di Pietro3, André Harnist3

1MOX, Politecnico di Milano, Italy; 2Université Côte d'Azur, Nice, France; 3IMAG, Université de Montpellier, France

We design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed discretization hinges on discontinuous polynomial unknowns on the mesh and on its skeleton, from which discrete differential operators are reconstructed. The reconstruction operators are then used to define a consistency term inspired by the weak formulation of the creeping flow problem and a cleverly designed stabilization term penalizing boundary residuals.The method has several appealing features including the support of general meshes and abitrary approximation order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray–Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau–Yasuda models. A key tool in our analysis is a generalization of the discrete Korn inequality to the non-Hilbertian setting. The theoretical results are demonstrated on a complete panel of numerical tests.



1:00pm - 1:30pm

Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order Methods

Théophile Chaumont-Frelet1,2, Alexandre Ern3,1, Simon Lemaire1,4, Frédéric Valentin5

1Inria; 2Laboratoire J.A. Dieudonné; 3CERMICS, Ecole des ponts; 4Laboratoire Paul-Painlevé; 5LNCC

We present and analyse two high-order, face-based, and multiscale schemes to approximate Darcy flow problems: the multiscale hybrid-mixed (MHM) and the multicale hybrid high-order (MsHHO) methods. The two methods hinge on element-wide PDE problems to define multiscale basis functions, leading in practice to a two-level divide-and-conquer strategy, since these local problems are uncoupled. The MHM and MsHHO methods have been recently introduced independently in the literature. Here, we show that even though the two methods seem very different at first sight, they actually produce the same numerical approximation when considering piecewise polynomial right-hand sides. Actually, we establish that the two methods employ the same discretization space, but rely on different sets of degrees of freedom that can be seen as dual to each other. To conclude, we propose improvements for the two schemes that are based on this observation.



 
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