International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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: 30th Nov 2022, 10:10:25pm CET
MS 14: efficient frameworks and implementations for multiphysics problems
The objective of this minisymposium is to present software frame-
works, key-ideas of their implementation as well as advanced discretisa-
tion techniques with applications to coupled and multiphysics problems.
In the recent years, a strong focus has been on the simulation of coupled
problems in fluid dynamics, elastodynamics and electrodynamics. Such
problems include, for instance, thermoelasticity, piezoelectricity, convec-
tion dominated flows or coupled flows with deformations in porous media,
or fluid-structure interactions involving moving domains, interface prob-
lems and crack propagation. To simulate multiphysics processes with
guaranteed stability and error control, key techniques like space-time fi-
nite element methods, space-time boundary element methods, high-order
(hybrid) discontinuous, mixed or unfitted finite element methods, as well
as matrix-free and multigrid methods, have been advanced. Due to their
involved structure, development and evaluation of efficient discretisations
and solvers require advanced software tools. The aim of the minisympo-
sium is to gather researchers from numerical mathematics and developers
of high-level software libraries and tools to foster scientific exchange for
the development of efficient and scaling software frameworks employing
12:00pm - 12:30pm
A variational localization approach for goal oriented adaptive refinement of parabolic PDEs
Leibniz University Hannover, Germany
In this talk we provide an approach for extending the PU-DWR (partition of unity, dual-weighted-residual) method
for goal-oriented error estimation to space-time Galerkin formulations.
This allows for the construction of a global error estimator w.r.t. a given goal functional,
but requires a Galerkin approach of the problem.
Space-time Galerkin (weak) formulations then allow for an estimation
of both spatial and temporal errors.
In the classical DWR method the element-wise error estimator is calculated using
integration by parts. This necessitates an evaluation of strong residuals and
jump terms across element boundaries, which can be difficult to implement.
We propose to use a partition-of-unity (PU) approach instead.
If we choose nodal basis functions we obtain a node-wise error contribution.
For stationary problems this approach yields good functional error convergence
and is relatively straightforward to implement.
Numerical results substantiate our developments.
12:30pm - 1:00pm
Adaptive space-time finite element methods for parabolic optimal control problems
Johannes Kepler University Linz, Austria
We present, analyze, and test locally stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard $L_2$-regularization. We derive a prior discretization error estimates in terms of the local mesh size. The adaptive version is driven by local residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in the form of a first expanding and then contracting ball in 3D that is fixed in the 4D space-time cylinder.
1:00pm - 1:30pm
Simplex Space-Time Meshes in Engineering Applications
1Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, United States of America; 2Chair for Computational Analysis of Technical Systems (CATS), RWTH Aachen University, Germany
The aim of this talk is to highlight how simplex space-time meshes can be used in production engineering applications within the framework of adaptive time-stepping, as already applied to benchmark cases in  and more realistic examples in . Their numerical simulation based on computational fluid dynamics is tremendously demanding and accounts for many physical phenomena, such as two-phase flows, phase transition, etc. Nevertheless, the efficiency of our simulations is improved when using adaptive temporal refinement in areas of interest.
A space-time finite element discretization, by means of 4D simplex space-time elements, referred to as pentatopes , leads to entirely unstructured grids with varying levels of refinement both in space and in time. We use a stabilized space-time finite element method in order to discretize the equations. The stabilization parameter is defined based on the contravariant metric tensor, as shown in . Its definition is extended in 4D and allows us to deal with complex anisotropic simplex meshes in the space-time domain. Furthermore, space-time elements are capable of connecting different spatial meshes at the bottom and top levels of the space-time slab and dealing with complicated domain movements/rotations that the standard ALE techniques cannot resolve without remeshing .
 V. Karyofylli, M. Frings, S. Elgeti and M. Behr. Simplex space-time meshes in two-phase flow simulations. International Journal for Numerical Methods in Fluids, 86(3):218-230, 2018.
 V. Karyofylli, L. Wendling, M. Make, N. Hosters and M. Behr Simplex space-time meshes in thermally coupled two-phase flow simulations of mold filling. Computers & Fluids, 192, 2019.
 M. Behr. Simplex space-time meshes in finite element simulations. International Journal for Numerical Methods in Fluids, 57(9):1421-1434, 2008.
 M. von Danwitz, V. Karyofylli, N. Hosters and M. Behr Simplex space-time meshes in compressible flow simulations. International Journal for Numerical Methods in Fluids, 91(1):29-48, 2019.
1:30pm - 2:00pm
Unfitted space-time finite element methods for PDEs on evolving geometries
1Institute of Numerical and Applied Mathematics, University of Göttingen, Germany; 2Max Planck Institute for Solar System Research, Göttingen Germany
The methodology of unfitted finite element methods, i.e. methods which are able to cope with interfaces or boundaries which are not aligned with the grid, have been investigated for different problems in recent years. However, the development of numerical methods which are flexible with respect to the geometrical configuration, robust and higher order accurate at the same time is still challenging. One major issue in the design and realization of higher order finite element methods is the problem of accurate and stable numerical integration on time-dependent (level set) domains. In this talk, we present a discontinuous-in-time space-time finite element method which allows for a higher order accurate and robust numerical treatment of domains that are prescribed by level set functions. To obtain higher order accuracy in space we use an approach that is based on specifically tailored isoparametric mappings. In order to handle ill-posed cut configurations, a Ghost penalty stabilisation is used. For illustration, the method is applied to a convection-diffusion problem and we mention theoretical results as well as numerical examples. Last, we review numerically what might be regarded as a continuous-in-time Petrov-Galerkin variant of the method.
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