International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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MS 29b: p- and hp-FEM and applications
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).
11:50am - 12:20pm
Multigrid solution strategies for immersed discretizations involving multi-level hp-refinement
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) .
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  is a high-order immersed
method for the numerical analysis of domains with a complex geometry while the multi-level hp-
method  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.
 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.
 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-
 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
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
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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