Abstracts:

Istvan Farago: On the AN-stability of one-step numerical methods

A-stable numerical methods are widely recommended for ordinary differential systems with a bounded and/or contractive numerical solution, as for Dahlquist's test equation $y'=\lambda y$ such methods produce a bounded and contractive numerical solution when $Re \lambda \leq 0$ by any choice of the time-step size. However, A-stable methods are not always able to produce a bounded/contractive numerical solution when the exact solution has these properties. A possible generalization called AN-stability, allowing a time-dependent coefficient $\lambda(t)$ was advocated by Butcher for Runge-Kutta methods. In this talk, we reformulate the definition of AN-stability for one-step methods and investigate its conditions about the implicit Euler method and its Richardson extrapolated version as well as for a special class of two-stage consistent Runge-Kutta methods.

Tom Gustafsson: Distributed finite elements using model order reduction

Cloud offers lots of compute resources but supercomputers have faster interconnectivity by default. We study a special overlapping finite element domain decomposition technique which eliminates the need for MPI so that a traditional supercomputer is not required. We present numerical results with up to 85 M degrees-of-freedom on a basic laptop by distributing the bulk of the model order reduction to low cost (i.e. low priority) cloud instances.

Jon Eivind Vatne: Angles and angle conditions for polytopes of high dimension

In a tetrahedron, one can consider the angles between edges on a triangular face, as well as the angles between the faces themselves; the latter are the dihedral angles. In any dimension, one can similarly consider angles of various types. In finite element methods, it is important to control the angles appearing in elements, in particular to ensure convergence. Partly motivated by this, we have considered a number of questions related to polytopes of higher dimensions, including the dihedral angle sums, angle conditions generalizing useful conditions from two and three dimensions, refinements of partitions preserving angle conditions, and special classes of polytopes that have particularly nice properties in this regard. Among the tools needed to analyze angles in higher dimensions is the generalized sine function studied by Swedish mathematician Folke Eriksson. This is a report on joint work with S. Korotov.

Tomáš Vejchodský: A posteriori error bounds for eigenfunctions

We consider an eigenvalue problem for a symmetric linear elliptic partial differential operator discretized by the usual conforming finite element method. We show how to generalize the Davis--Kahan theorem to
the weakly formulated eigenvalue problems and how to combine it with the Prager--Synge technique to obtain highly accurate and fully computable a
posteriori error bounds on eigenfunctions. This approach applies well even in the case of multiple eigenvalues and tight clusters.

Jean-Paul Murara: A numerical solution of the investment-consumption optimal control problem

In this talk, I will introduce a modification of the Merton's investment-consumption problem. Considering a non-constant volatility environment, we construct a more realistic portfolio based on new assumptions. A related stochastic optimal control problem and its corresponding Hamilton-Jacobi-Bellman equation are derived.  A numerical analysis is then performed in order to obtain the optimal solution.

János Karátson: Streamline diffusion preconditioning of Krylov iterations for convection-dominated elliptic problems

Convection-dominated elliptic equations form an important class in the modelling of stationary convection-diffusion problems, for which a popular numerical solution strategy is the the streamline diffusion finite element method (SDFEM). This talk is devoted to the SDFEM combined with equivalent preconditioning in the iterative solution of the arising algebraic systems. The preconditioner is obtained from the streamline diffusion inner product. First it is proved via a streamline version of Poincaré-Friedrichs inequality that the linear convergence estimation is robust, i.e.  bounded independently of the perturbation parameter, for proper convection vector fields. Then we study the superlinear phase of convergence, and provide some estimations for the limiting case on the operator level.

Sergey Korotov: FEM may converge on very bad meshes

In this talk I will discuss some geometric conditions on meshes related to the convergence of (classical) finite element approximations.

Jakub Šístek: Towards a scalable domain decomposition solver for immersed boundary finite element method

Immersed boundary finite element method (FEM) presents an attractive approach to simulations avoiding the generation of large body-fitted meshes. This can be tedious and challenging for complex geometries as well as for very fine adaptive meshes distributed over a parallel supercomputer. However, the price to pay are more complicated formulations for the weak enforcement of Dirichlet boundary conditions, poor conditioning of stiffness matrices, and nonstandard numerical integration at the boundary. We develop multilevel balancing domain decomposition based on constraints (BDDC) method tailored to the solution of the linear systems arising in the context of immersed boundary FEM with parallel adaptive grid refinement using Z-curves (based on the p4est library). One crucial challenge is presented by fragmenting of subdomains, which has two sources: i) the partitioning strategy based on space-filling curves, and ii) extraction of the elements contributing to the stiffness matrix. Numerical results for large-scale Poisson and linear elasticity problems on complex geometries from engineering will be presented.
This is joint work with Monika Balázsová, Fehmi Cirak, Eky Febrianto, Matija Kecman, Pavel Kůs, and Josef Musil.

Maksat Ashyraliyev: Stable difference schemes for the solution of the source identification problem for telegraph-parabolic equations

We construct first- and second-order accurate difference schemes for the approximate solution of the inverse problem for telegraph-parabolic equations with an unknown spacewise-dependent source term. The unique solvability of the constructed difference schemes and the stability estimates for their solutions are obtained. We discuss the numerical procedure for the implementation of these difference schemes and provide an error analysis for the numerical results of simple test problems.

Bartosz Malman: A short introduction to the Crouzeix conjecture

The work of the numerical analyst Michel Crouzeix from 2004 has over the last two decades caught the interest of many operator theorists, and his conjecture occupies a cental place in modern theory. If p is a polynomial, and we know the maximum of this polynomial over the numerical range of a matrix A, then what can be said about the operator norm of p(A)? The famous Crouzeix conjecture claims that the operator norm is at most twice this maximum. In my talk, I will introduce the conjecture and related results, as well as describe my own work on the subject, obtained in a collaboration with Ryan O'Loughlin, Javad Mashreghi and Thomas Ransford.