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.