Domain decomposition preconditioning for the Helmholtz equation: a coarse space based on local Dirichlet-to-Neumann maps

In this thesis, we present a two-level domain decomposition method for the iterative solution of the heterogeneous Helmholtz equation. The Helmholtz equation governs wave propagation and scattering phenomena arising in a wide range of engineering applications. Its discretization with piecewise linear finite elements results in typically large, ill-conditioned, indefinite, and non- Hermitian linear systems of equations, for which standard iterative and direct methods encounter convergence problems. Therefore, especially designed methods are needed. The inherently parallel domain decomposition methods constitute a promising class of preconditioners, as they subdivide the large problems into smaller subproblems and are hence able to cope with many degrees of freedom. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be fatal. We develop a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. Apart from the question of how to design the coarse space, we also investigate the question of how to incorporate the coarse space into the method. Also here the fact that the stiffness matrix is non-Hermitian and indefinite constitutes a major challenge. The resulting method is parallel by design and its efficiency is investigated for two- and three-dimensional homogeneous and heterogeneous numerical examples

Addendum to "A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator"' [J. Comput. Appl. Math. 271 (2014) 83–99]

This communication gives an addendum to the paper Conen et al. (2014)

A first update on mapping the human genetic architecture of COVID-19

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FEM/FD Immersed Boundary FSI Simulations

Immersed boundary simulations have been under development for physiological flows, allowing for elegant handling of fluid-structure interaction modelling with large deformations due to retained domain-specific meshing. We couple a structural system in Lagrangian representation that is formulated in a weak form with a Navier-Stokes system discretized through a finite differences scheme. We build upon a proven highly scalable imcompressible flow solver that we extend to handle FSI. We aim at applying our method to investigating the hemodynamics of Aortic Valves. The code is going to be extended to conform to the new hybrid-node supercomputers

A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be devastating. In this paper, we present a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples

2013 ESH/ESC Guidelines for the management of arterial hypertension: The Task Force for the management of arterial hypertension of the European Society of Hypertension (ESH) and of the European Society of Cardiology (ESC).

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