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: 8th Dec 2022, 11:35:37pm CET

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Session Overview
MS 4a: high order methods on polyhedral meshes
Monday, 12/July/2021:
1:50pm - 3:50pm

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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1:50pm - 2:20pm

Polytopic Discontinuous Galerkin Methods for Radiation Transport Problems

Paul Houston

University of Nottingham, United Kingdom

In this talk we develop discontinuous Galerkin finite element methods for the discretization of the radiation transport problem on general (spatial) computational meshes consisting of polygonal/polyhedral (polytopic) elements. Our particular interest is the application to medical treatment planning in clinical radiotherapy. Here we study both the stability and a priori error analysis of the proposed scheme. The implementation is based on exploiting a nodal approximation in energy and angle, together with fast numerical integration techniques on the spatial polytopic mesh; this approach leads to a highly parallelisable algorithm whereby a large number of linear transport solves must be computed. Numerical experiments are presented to highlight the accuracy of the proposed method, as well as to benchmark with more standard kinetic Monte Carlo simulations.

2:20pm - 2:50pm

Convection-robust SUPG Virtual Elements

Lourenco Beirao da Veiga1, Franco Dassi1, Carlo Lovadina2, Giuseppe Vacca1

1Università di Milano-Bicocca, Italy; 2Università di Milano, Italy

The focus of this talk is describing a SUPG-stabilized Virtual Element Method of general ``polynomial'' order for diffusion-convection problems, that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show, for the first time in the literature, an “almost uniform” error bound (in the sense that the unique term that depends in an unfavorable way on the parameters is damped by a higher order mesh-size multiplicative factor). Furthermore, we introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We will also present some numerical result in accordance with the theoretical developements. If time allows, we will also show a glimpse of a divergence-free VEM SUPG scheme for the Oseen problem.

2:50pm - 3:20pm

A high-order discontinuous Galerkin method on polygonal and polyhedral grids for the poro-elasto-acoustic problem

Paola F. Antonietti1, Michele Botti2, Ilario Mazzieri3, Simone Nati Poltri4

1Politecnico di Milano, Italy; 2Politecnico di Milano, Italy; 3Politecnico di Milano, Italy; 4Politecnico di Milano, Italy

The present talk deals with the numerical analysis of the coupled poro-elasto-acoustic differential problem modelling acoustic/sound wave impacting a poroelastic medium and consequently propagating through it. Wave propagation is modelled by the acoustics equations in the acoustic domain and by the low-frequency Biot's equations in the poroelastic one. The coupling is realised by means of (physically consistent) transmission conditions, imposed on the interface between the domains, modelling different pore configurations. For the space discretisation, we introduce and analyse a high-order discontinuous Galerkin method on polygonal and polyhedral meshes, which is then coupled with Newmark-beta time integration schemes. Stability analysis for both the continuous and semi-discrete problem is presented and error estimates in a suitable energy norm are derived for the semi-discrete formulation. Numerical results obtained on test cases with manufactured solutions are presented in order to demonstrate the theoretical error bounds. Examples of physical interest are also presented to investigate the capability of the proposed methods in practical scenarios.

3:20pm - 3:50pm

Mesh adaptivity for polygonal and polyhedral meshes in flow simulations through fractured media

Stefano Berrone, Andrea Borio, Alessandro D'Auria, Fabio Vicini

Politecnico di Torino, Dipartimento di Scienze Matematiche, Italy

Flow simulations in fractured media are typically characterized by a strong geometrical complexity and by non-smooth and strongly localised solutions. Within this framework several issues concerning efficient large scale simulations raises, among them we can list: construction of a good quality mesh fitting the geometry, construction of a mesh suitable for the representation of the solution and its singularities, reliability and efficiency of the numerical methods used in the simulations. Reliability of the solution is a key issue due to environmental impact assessments and the huge costs of the human activities operating in this field. Efficiency is connected with the uncertainty of the geometry and problems parameters requiring many simulations for Uncertainty Quantification of the quantities of interest.

In this talk several of this topics will be tackled focusing on few of them related to mesh adaptivity and "medium" order methods.

Polygonal and polyhedral methods allow the introduction of a natural strategy to obtain a coarse mesh conforming to the geometrical inclusions or domain intersections where it is known the singularities of the solution lie. The quality of this mesh is usually very bad and even worse in 3D problems. Based on a posteriori error estimates the refinement process aims at improving the quality of the mesh while improving the representation of the solution. Due to the limited regularity of the solutions to the problem considered and the presence in the mesh of odd-elements tests are performed with virtual elements of moderate order rather than high order.

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