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 29b: p- and hp-FEM and applications
Time:
Wednesday, 14/July/2021:
11:50am - 1:50pm

Session Chair: Lothar Banz
Session Chair: Andreas Schröder
Virtual location: Zoom 3


Session Abstract

The aim of this minisymposium is to discuss recent developments and applications of p- and hp-methods

in the context of finite elements, boundary element methods and discontinuous Galerkin methods. It is

very well known that these methods have high convergence properties in terms of the degrees of free-

dom, but they often require specific adaptations in order to ensure their efficiency in terms of computa-

tional costs or their applicability to involved, possibly nonlinear problems. The minisymposium addresses

several topics of p- and hp-methods with a certain focus on algorithmic aspects and covers, for instance,

hp-adaptivity based on error control, numerical integration, fast solvers, implementation aspects (e.g.

evaluation of element matrices) and applications to real world problems (e.g. image based simulations

with more than one billion degrees of freedom).


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Presentations
11:50am - 12:20pm

Multigrid solution strategies for immersed discretizations involving multi-level hp-refinement

Stefan Kollmannsberger, John Jomo, Oguz Oztoprak, ernst rank

Technische Universität München, Germany

Multigrid methods use a hierarchy of coarse discretizations to accelerate the convergence of large

systems. They are characterized by convergence rates that are independent of the mesh size and

yield efficient iterative schemes that are well-suited for large-scale finite element analysis on parallel

machines when used either as a stand-alone solver or a preconditioner within a Krylov method.

In hp-finite element methods, it is possible to define the sequence of coarse discretizations in a

multigrid algorithm based on meshes with varying element sizes or refinement levels (h-multigrid),

polynomial orders (p-multigrid) or both (hp-multigrid) [1].

Our contribution presents an efficient hp-multigrid method for the solution of large finite cell

systems involving multi-level hp-refinement. The finite cell method [2] is a high-order immersed

method for the numerical analysis of domains with a complex geometry while the multi-level hp-

method [3] is an efficient hp-scheme based on refinement by superposition. We take advantage of the

hierachic basis functions used in both schemes and the superposition principle used in the multi-level

hp-method to develop a simple and elegant multigrid framework for the solution of immersed problems

in the range of multiple millions and billions of unknowns. This approach utilizes smoothers that are

robust with respect to cut cells resulting in convergence that is independent of the cut configuration,

the mesh size and in special cases even the polynomial order.

The structure and performance of the multigrid framework as well as the smoothers needed to deal

with conditioning problems related to cut-cells will be shown.

References

[1] Graig, A. W. and Zienkiewicz, O. C. A multigrid algorithm using a hierarchical finite element

basis. In Multigrid Methods for Integral and Differential Equations. Clarendon Press, 1985.

[2] Düster, A. and Parvizian, J. and Yang, Z. and Rank, E. The finite cell method for three-

dimensional problems of solid mechanics. Computer Methods in Applied Mechanics and Engi-

neering 2008.

[3] Zander, N. and Bog, T. and Kollmannsberger, S. and Schillinger, D. and Rank E. Multi-level

hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes.

Computational Mechanics 2015.



12:20pm - 12:50pm

hp Galerkin BEM --- exact quadrature on n-dim polyhedral domains

Matthias Maischak

Brunel University London, United Kingdom

We show a fully analytic quadrature method for the n-dimensional Galerkin boundary element method for all standard kernels and arbitrary polynomial degrees on generalized elements.

Generalized elements are polyhedral subsets with planar faces.

Our algorithmic approach minimizes the number of integrals to solve and puts the emphasis on the algebraic construction.

In addition to recurrence formulas we derive a closed form sum representation.

Numerical results for some selected examples will be presented which illustrate the feasibility of the exact quadrature method.



12:50pm - 1:20pm

Symbolic evaluation of hp-FEM element matrices

Tim Haubold1, Sven Beuchler1,3, Veronika Pillwein2

1Leibniz University Hannover, Germany; 2Johannes Kepler University Linz, Austria; 3Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering - Innovation Across Disciplines), Leibniz Universität Hannover, Germany

In this talk we consider high-order finite element discretizations of linear elliptic boundary value problems.

Following, e.g. Beuchler et al., 2012 or Karniadakis, Sherwin 1999, a set of hierarchic basis functions on simplices is chosen. For an affine simplicial triangulation this leads to a sparse stiffness matrix. Moreover the L^2-inner product of the interior basis functions is sparse with respect to the polynomial order p. The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials.

In this talk we present algorithms which compute the remaining non zero entries of mass- and stiffness matrix in optimal arithmetical complexity. In order to obtain this result, recursion fomulas based on symbolic methods, see e.g. Kauers: SumCracker - A Package for Manipulating Symbolic Sums and Related Onjects, (2006), are used.

The presented techniques can be applied not only to scalar elliptic problems in H^1 but also for vector valued problems in H(div) and H(curl) where an explicit splitting of

the higher-order basis functions into solenoidal and non-solenoidal ones is used.



 
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