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.
David KayUniversity of Oxford
Numerical Methods for Fractional Diffusion
Fractional differential equations are becoming increasingly used as a modelling tool for coping with anomalous diffusing processes or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues and this imposes a number of computational constraints. In this talk we develop efficient, scalable techniques for solving fractional-in-space reaction-diffusion equations using both finite element and finite difference methods. In the case of finite elements we present robust techniques for computing the fractional power of a Laplacian matrix times a vector. Thereby overcoming the issues of non-locality inherent within the problem. In the finite difference case the application and analysis of multigrid methods for the resulting matrices are considered. Finally, we discuss the advantages and downfalls of both methods. Numerical results show case the methods by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions.
Dr. Fehmi CirakUniversity of Cambridge
Title: Subdivision-stabilised Immersed B-Spline Finite Elements for Fluid-Structure Interaction
Abstract: The computation of systems involving the two-way interaction between fluids and lightweight structures is rife with challenges due to differences in the underlying equations and the disparity of the length and time scales involved. In this talk, a new immersed finite element method is introduced for computing fluid-structure interaction problems with geometrically and topologically complex interfaces. The viscous, incompressible fluid is discretized using a fixed Cartesian grid and b-spline basis functions. The two-scale relationship of b-splines is utilised to implement an intriguingly simple and efficient technique to satisfy the LBB condition. On non-grid-aligned fluid domains and at moving fluid-structure interfaces, the boundary conditions are enforced with a consistent penalty method as originally proposed by Nitsche. In addition, a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut-cells. In contrast to the fluid, the structure is represented by beams, membranes or thin shells and is discretized with subdivision finite elements. The interaction between the fluid and structure is accomplished by means of a strongly coupled iteration scheme.