Konstatinos ZygalakisUniversity of Southampton & EPFL
Qualitative Behaviour of Numerical Methods for SDEs and Applications
Patrick LamUniversity of Waterloo, Canada
‘Views, Programming Languages Research, and HPC’
Jean-Francois RemacleUniversité catholique de Louvain
Title: Generation of probably correct curvilinear meshes
Abstract: There is a growing consensus that state of the art Finite Volume technology requires, and will continue to require too extensive computational resources to provide the necessary resolution, even at the rate that computational power increases. The requirement for high resolution naturally leads us to consider methods which have a higher order of grid convergence than the classical (formal) 2nd order provided by most industrial grade codes. This indicates that higher-order discretization methods will replace at some point the finite volume solvers of today, at least for part of their applications.
The development of high-order numerical technologies for CFD is underway for many years now.
For example, Discontinuous Galerkin methods (DGM) have been largely studied in the literature, initially in a quite theoretical context, and now in the application point of view. In many contributions, it is shown that the accuracy of the method strongly depends of the accuracy of the geometrical discretization. In other words, the following question is raised: it is true that we have the high order methods, but how do we get the meshes?
Timo BetckeUniversity College London
Title : Spectral decompositions and non-normality of boundary integral operators in acoustic scattering.
Abstract: Spectral properties of boundary integral operators in acoustic scattering are essential to understand properties such as coercivity or convergence of iterative methods. Yet, very little is known apart from the unit disk case, where the Green's function has a simple decomposition into Fourier modes. For more general domains it is not even known whether boundary integral operators are normal. In this talk we first extend known results about eigenvalues on the circle to the case of an ellipse, where a decomposition of the acoustic Green's function in elliptic coordinates is possible. Based on this it is shown that scaled versions of the standard boundary integral operators are normal in a modified $L^2$ inner product on the ellipse. For more general domains we will define approximate decompositions with fast decaying errors, that are normal in a scaled L^2 inner product and thereby demonstrate that boundary integral operators are in some sense close to normal operators also on general domains. We will present numerical examples on several interesting domains that show how non-normality influences the properties and the numerical behavior of boundary integral operators.