Program:
13:30 – 14:00, Carl Lundholm, Umeå university
A Space-Time CutFEM on Overlapping Meshes
Abstract: We present a cut finite element method for the heat equation on two overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a stationary background mesh at the bottom and an overlapping mesh that is allowed to move around on top of the background mesh. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing geometry. In this work the overlapping mesh is prescribed a simple continuous mesh movement, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method and also includes an integral term over the space-time boundary that mimics the standard discontinuous Galerkin time-jump term. The continuous mesh movement results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore propose a new energy analysis framework that is general and robust enough to be applicable to the current setting. The main result of the energy analysis is an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.
14:30 – 15:00, Niklas Lundström, Umeå university
Estimates of size of cycle in a predator-prey system
Abstract: We consider a standard Rosenzweig–MacArthur predator-prey system, assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which should be of independent interest. This is joint work with Gunnar Söderbacka.
15:30 – 16:00, Eric Libby, Umeå university
Metabolic compatibility and the rarity of prokaryote endosymbioses
Abstract: Endosymbiosis is extremely rare in bacteria and archaea. Yet one such endosymbiosis gave rise to eukaryotes and eventually complex multicellular life. While many factors may contribute to the rarity, we lack ways to estimate their influence. Here, we develop a quantitative approach to estimate the influence of one possible factor: metabolic compatibility, which is the ability of both host and endosymbiont to grow on the same resources. Using metabolic networks of existing prokaryotes we find that more than half of possible endosymbioses would be metabolically viable; however, the resulting endosymbioses are less fit than their ancestors and unlikely to gain adaptations that overcome their fitness disadvantages. Our results provide null models that shed light on the diversity of prokaryotic life.
