European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS21: Structure-preserving discretization methods I: Discretization methods based on exterior calculus (Part 3)
Chair: Marc Gerritsma
08:30
25 mins
A conservative n-level hybrid scheme for high order methods
Varun Jain, Yi Zhang, Artur Palha, Marc Gerritsma
Abstract: Hybrid methods are classical domain decomposition methods that first solve for interface Lagrange multipliers between the elements. These interface degrees of freedom are then used as boundary conditions to solve for solution within each element separately. The interface problem is a reduced problem as compared to the full global system [1]. However, for large scale problems the interface system in itself can become too large. In this work we show that the reduced interface system is in itself a hybrid system that can also be solved efficiently – as in the case of any hybrid system. Furthermore, the reduced system of the interface system is in itself a hybrid system. In general, the reduced system of a hybrid system is also a hybrid system. Therefore, it is possible to have n-level hybridization of the scheme that reduce the problem of solving one big system to solve for n-hybrid systems that can solved very efficiently and easily parallelized. We will illustrate this scheme with a numerical example on mixed formulation of Poisson equation using high order mimetic spectral element method. [1] Jain, V., Zhang Y., Palha A., Gerritsma M., A conservative hybrid method for Darcy flow, submitted to ICOSAHOM, 2018.
08:55
25 mins
A HYBRID SPECTRAL EQUILIBRIUM FORMULATION FOR ELASTICITY PROBLEMS
Marc Gerritsma, Joel Fisser, Varun Jain, Yi Zhang
Abstract: In this presentation, a hybrid equilibrium spectral element formulation will be presented which point wise conserves linear and angular momentum. This is a mixed formulation in which we solve for the stress and displacement field independently. The numerical error stems from the approximation of the constitutive law which links the stresses to the deformation rate. The hybrid formulation decouples the solution between elements which are then explicitly coupled again through the use of suitably chosen Lagrange multipliers. By means of a blend of primal and dual representations, the well-known spurious kinematic modes, which tend to pollute solutions in hybrid formulations, are avoided. Furthermore, this formulation allows to solve for the trace variables first after which the problem decouples into independent problems which can be solved in parallel.
09:20
25 mins
Variational discretisations for wave-ship and wave-energy dynamics
Onno Bokhove
Abstract: Enumath2019, Egmond aan Zee, The Netherlands Variational discretisations for wave-ship and wave-energy dynamics By Onno Bokhove, School of Mathematics, University of Leeds, Leeds, UK o.bokhove@leeds.ac.uk Variational methods are investigated analytically and numerically to model (nonlinear) water waves interacting with structures such as ships and a novel wave-energy device. In particular, we will discuss discontinuous Galerkin finite-element time integrators and constrained continuous Galerkin finite-element dynamics in space. As an illustration/validation, our modelling results using (dis)continuous Galerkin finite element methods will be developed for a novel wave-energy device with full coupling between the water waves, the constrained buoy motion and the electromagnetic actuator. References O. Bokhove and A. Kalogirou 2016: Variational Water Wave Modelling: from Continuum to Experiment. Editors: Bridges, Groves and Nicholls. London Mathematical Society Lecture Notes Series 426, 226-259. O. Bokhove, A. Kalogirou and W. Zweers 2019: From bore-soliton-splash to rogue waves, a new wave-energy device and extreme tsunami run-up. Submitted to Water Waves. https://eartharxiv.org/5p8un