15:45
MS21: Structure-preserving discretization methods I: Discretization methods based on exterior calculus (Part 2)
Chair: Marc Gerritsma
15:45
25 mins
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Structure Preserving Methods for the Vlasov-Maxwell equations based on a variational principle
Eric Sonnendrücker
Abstract: Conservation laws in physics models can often be related to symmetries related to a variational principle. The aim of this talk is to highlight how a discrete version of the variational principle can be constructed using a structure preserving discretisation of the physical quantities appearing in the variational principle. Such a discretisation can be constructed based on different techniques like Finite Differences, Finite Elements or spectral methods.
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16:10
25 mins
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High-Order Discretizations of Scalar Curvature and Geometric Evolution Equations with Regge Finite Elements
Evan Gawlik
Abstract: A widely used approximation of the Gaussian curvature on a triangulated surface is the angle defect, which measures the deviation between $2\pi$ and the sum of the angles between neighboring edges emanating from a common vertex. We show that the linearization of the angle defect about an arbitrary piecewise constant metric (more precisely, a piecewise constant Regge metric) is related to the classical Hellan-Herrmann-Johnson finite element discretization of the div-div operator. Integrating this relation leads to an integral formula for the angle defect which is well-suited for analysis and generalizes naturally to higher order. We prove error estimates for these high-order approximations of the Gaussian curvature in $H^k$-Sobolev norms of integer order $k \ge -1$.
A central role in our work is played by the Regge finite element spaces recently developed by Li, which have their origins in the work of Christiansen. These finite element spaces consist of piecewise polynomial $(0,2)$-tensor fields with continuous tangential-tangential components across element interfaces. In the lowest order setting, a positive definite Regge finite element realizes a piecewise flat triangulation whose (squared) edge lengths correspond to the degrees of freedom for the finite element space.
After constructing high-order scalar curvature approximations, we use them to derive finite element methods for the solution of evolution equations in Riemannian geometry. Our focus is on Ricci flow and Ricci-DeTurck flow in two dimensions. In the lowest order setting, the finite element method we develop for two-dimensional Ricci flow is closely connected with a popular discretization of Ricci flow in which the scalar curvature is approximated with the angle defect. We present some results from our ongoing work on the analysis of the method, and we conclude with numerical examples.
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16:35
25 mins
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Algebraic flux correction for advection of scalar and symmetric tensor fields
Christoph Lohmann
Abstract: The work to be presented in this talk extends the algebraic flux correction (AFC) methodology to advection(-reaction) equations and symmetric tensor fields \cite{lohmann2019b}. The new theoretical results are used to design bound-preserving finite element methods for steady and unsteady model problems. The proposed approaches add algebraically defined artificial diffusion operators to the Galerkin discretization. Then limited antidiffusive fluxes are incorporated into the residual of the resulting low order method to remove redundant diffusivity and to improve the accuracy of the approximation.
In the case of the steady state advection problem with a scalar solution, convergence with order $\frac 12$ is shown by adapting an a priori error estimate derived in \cite{Barr2016}. Existence of a unique solution is proved under suitable assumptions. Furthermore, sufficient conditions for the validity of generalized discrete maximum principles (DMPs) are formulated. In addition to guaranteeing boundedness of the function values in terms of weakly imposed boundary conditions, they provide local $L^\infty$ estimates for subsets of degrees of freedom. These DMP results are extended to the transient advection equation discretized in time by the $\theta$-scheme. A priori time step restrictions are derived for $\theta \in [0,1)$. The analysis of the forward Euler time discretization implies that bound-preserving approximations of higher order can be obtained without solving nonlinear systems if explicit strong stability preserving (SSP) Runge-Kutta time integrators are employed.
Based on the results of theoretical studies, new definitions of correction factors are proposed which potentially facilitate the development of more efficient solvers and/or implementations.
Using the concept of L\"owner ordering, the scalar AFC framework is extended to the numerical treatment of symmetric tensor quantities. In tensorial versions of the methods under consideration, antidiffusive fluxes are limited to constrain the eigenvalue range of the tensor field by imposing discrete maximum principles on the extremal eigenvalues \cite{lohmann2018}. This criterion is shown to be an appropriate frame invariant generalization of scalar maximum principles. It leads to a family of robust property-preserving limiters based on the same design principles as their scalar counterparts.
The potential of the presented methods is illustrated by numerical examples.
\begin{thebibliography}{10}
\bibitem{Barr2016} G.~R.~Barrenechea, V.~John, and P.~Knobloch,
``Analysis of Algebraic Flux Correction Schemes'',
\textit{SIAM Journal on Numerical Analysis} 54.4, pp. 2427--2451, 2016.
\bibitem{lohmann2018} C.~Lohmann,
``Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields'',
\textit{ESAIM: M2AN}, (in press, online version available at
https://doi.org/10.1051/m2an/2019006).
\bibitem{lohmann2019b} C.~Lohmann,
``Physics-compatible finite element methods for scalar and tensorial advection problems'',
\textit{PhD thesis}, TU Dortmund University, 2019.
\end{thebibliography}
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