ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
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

Session Overview 
Session  
MS 4b: high order methods on polyhedral meshes
 
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 subsurface porous media flow, to elastic deformation of materials, to fluidstructure 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.  
Presentations  
11:50am  12:20pm
Arbitraryorder fully discrete discrete de Rham complex on polyhedral meshes ^{1}IMAG, Univ Montpellier, CNRS, Montpellier, France; ^{2}School 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 nonconforming 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) L2norm should be controlled by the (discrete) L2norm 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 nonconforming settings, to verify global integration by parts formulas exactly. In this work we present a novel arbitraryorder 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
hpVERSION DISCONTINUOUS GALERKIN METHODS ON ESSENTIALLY ARBITRARILYSHAPED ELEMENTS ^{1}SISSA, Italy; ^{2}INRIA, France; ^{3}University of Leicester, UK; ^{4}NTUA, Greece We extend the applicability of InteriorPenalty (IP) discontinuous Galerkin (dG) methods for advectiondiffusionreaction 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 H1L2type inverse estimate enables the proof of infsup stability of the method in a streamlinediffusionlike 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. hpVersion discontinuous Galerkin methods on essentially arbitrarilyshaped elements. ArXiv (2019):1906.01715. 12:50pm  1:20pm
Two error estimators for the p and hp versions of the virtual element method ^{1}University of Vienna, Austria; ^{2}University of Milano Bicocca, Italy; ^{3}University of Bonn, Germany; ^{4}Los 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 ^{1}University of Oldenburg, Germany; ^{2}ETH Zurich; ^{3}University of Vienna We develop the p and hpversions 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 Poissonlike and Stokeslike VE spaces for the velocities. This allows us to reinterpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poissonlike 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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