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).

Please note that all times are shown in the time zone of the conference. The current conference time is: 1st Dec 2022, 08:05:25am CET

 
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Session Overview
Session
MS 4b: high order methods on polyhedral meshes
Time:
Wednesday, 14/July/2021:
11:50am - 1:50pm

Session Chair: Claudio Canuto
Session Chair: Marco Verani
Virtual location: Zoom 1


Session Abstract

Polytopal Element Methods (PoEMs) are a collection of numerical methods used to compute ap-

proximate solutions to partial differential equations modelling a wide variety of physical phenomena,

from sub-surface porous media flow, to elastic deformation of materials, to fluid-structure interac-

tion problems. The novelty and recent interest in PoEMs stems from their ability to combine high

order discretization techniques with tessellation of physical domains using not only standard shapes

(triangles, tetrahedra, squares, cubes, etc) but also highly irregular geometries (arbitrary convex

polyhedra, unions of distinct shapes, specialized shapes for corners of a domain, etc).

This minisymposium brings together several specialists with complementary expertise in this excit-

ing and timely area of research to discuss the current status of the field and prospects for the future.


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Presentations
11:50am - 12:20pm

Arbitrary-order fully discrete discrete de Rham complex on polyhedral meshes

Daniele Antonio Di Pietro1, Jérôme Droniou2

1IMAG, Univ Montpellier, CNRS, Montpellier, France; 2School of Mathematics, Monash University, Melbourne, Australia

The design of stable and convergent schemes for the numerical approximation of certain classes of partial differential equations (PDEs) requires to reproduce, at the discrete level, the underlying geometric, topological, and algebraic structures. This leads to the notion of compatibility, which can be achieved either in a conforming or non-conforming setting. Relevant examples include PDEs that relate to the de Rham complex.

In order to serve as a basis for the numerical approximation of PDEs, discrete counterparts of this sequence of spaces and operators should enjoy the following key properties:

(P1) Complex and exactness properties. For the sequence to form a complex, the image of each discrete vector calculus operator should be contained in the kernel of the next one. Moreover, exactness properties depending on the topology of the domain should be reproduced at the discrete level.

(P2) Uniform Poincaré inequalities. Whenever a function from a space in the sequence lies in some orthogonal complement of the kernel of the vector calculus operator defined on this space, its (discrete) L2-norm should be controlled by the (discrete) L2-norm of the operator up to a multiplicative constant independent of the mesh size.

(P3) Primal and adjoint consistency. The discrete vector calculus operators should satisfy appropriate commutation properties with the interpolators and their continuous counterparts. Additionally, these operators along with the corresponding (scalar or vector) potentials should approximate smooth fields with sufficient accuracy. Finally, whenever a formal integration by parts is used in the weak formulation of the problem at hand, the vector calculus operators should also enjoy suitable adjoint consistency properties. The notion of adjoint consistency accounts for the failure, in non-conforming settings, to verify global integration by parts formulas exactly.

In this work we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into the ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts. The complete set of properties (P1)-(P3) has been recently established for this DDR complex. We also show how this DDR complex can be used for the practical analysis and design of approximation schemes.



12:20pm - 12:50pm

hp-VERSION DISCONTINUOUS GALERKIN METHODS ON ESSENTIALLY ARBITRARILY-SHAPED ELEMENTS

Andrea Cangiani1, Zhaonan Dong2, Emmanuil Georgoulis3,4

1SISSA, Italy; 2INRIA, France; 3University of Leicester, UK; 4NTUA, Greece

We extend the applicability of Interior-Penalty (IP) discontinuous Galerkin (dG) methods for advection-diffusion-reaction problems to meshes comprising extremely general curved element shapes. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace inverse estimates to Lipschitz element shapes. A further new H1-L2-type inverse estimate enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach. Time permitting, current work on a posteriori error estimates will also be discussed.

A. Cangiani, Z. Dong, E.H. Georgoulis. hp-Version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. ArXiv (2019):1906.01715.



12:50pm - 1:20pm

Two error estimators for the p- and hp- versions of the virtual element method

Lorenzo Mascotto1, Lourenco Beirao da Veiga2, Franco Dassi2, Joscha Gedicke3, Gianmarco Manzini4

1University of Vienna, Austria; 2University of Milano Bicocca, Italy; 3University of Bonn, Germany; 4Los Alamos National Laboratory, USA

We present two different error estimators for the p- and hp- versions of the virtual element method. The first one is a residual error estimator that presents the usual suboptimality in terms of the polynomial degree in the efficiency. The second one is an error estimator based on the hypercircle of Prager and Synge, which is reliable and efficient. Time permitting, in the latter case, we discuss difficulties in designing local flux reconstruction techniques in the virtual element setting.



1:20pm - 1:50pm

p- and hp- Virtual Elements for the Stokes problem

Alexey Chernov1, Carlo Marcati2, Lorenzo Mascotto3

1University of Oldenburg, Germany; 2ETH Zurich; 3University of Vienna

We develop the p- and hp-versions of the Virtual Element Method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. Thus we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. In this talk we touch upon the key steps in the a priori convergence analysis and demonstrate the performance of the method in several numerical examples.

Reference:

A. Chernov, C. Marcati, L. Mascotto, "p - and hp - virtual elements for the Stokes problem", Adv. Comput. Math. (2021), in press



 
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