The finite element method (FEM) is a fundamental tool of the numerical solution of real-life problems based on partial differential equations. In the recent decades, various generalizations of the standard FEM have been developed. A lot of such extensions have been motivated by difficulties, arising in physical or engineering problems, that may be cumbersome to overcome with standard FEM techniques. Such situations are the presence of boundary layers, singularities or discontinuities in the solution, complex and/or evolving geometry of the physical domains etc. The tools of extension of the FEM may be enriching the polynomial approximation space with non-polynomial shape functions, allowing general polygonal/polyhedral cells, or use a boundary-unfitted mesh and restricted shape functions (either to a bulk domain or to a surface). This survey type talk gives a brief introduction to the main ideas of some generalized FEMs that use the above ideas: XFEM, VEM, CutFEM and TraceFEM.
Sok szeretettel várunk minden kedves érdeklődőt a Budapesti Műszaki és Gazdaságtudományi Egyetem Természettudományi Karának Nyílt Napjára 2018. november 16-án (pénteken) a BME dísztermében.
Részletes információk: http://felvi.physics.bme.hu/nyiltnap
The talk summarizes our results obtained when studying the possible application of innovative integrators (operator splitting procedures and exponential integrators) to optimal control problems. We introduce the innovative integrators considered, and present the advantages and drawbacks of their use. After introducing the abstract optimal control problem having a linear quadratic cost function, we prove the convergence of innovative integrators applied to it. To do so, we briefly summarize the main results in operator semigroup theory, used in the proof. As an example, we treat the linear quadratic regulator problem of the one-dimensional and two-dimensional shallow water equations, and illustrate our results by numerical experiments.
The results are based on the joint works with Johannes Winckler (Tübingen) and Hermann Mena (Innsbruck).
For a certain type of discrete--time nonlinear consensus dynamics, asymptotically stable periodic orbits are constructed. Based on a simple ordinal pattern assumption, the Frucht graph, two Petersen septets, hypercubes, a technical class of circulant graphs (containing Paley graphs of prime order), and complete graphs are considered -- they are all carrying moving average monotone dynamics admitting asymptotically stable periodic orbits with period 2. Carried by a directed graph with 594 (multiple and multiple loop) edges on 3 vertices, also the existence of asymptotically stable r-periodic orbits, r=3,4,... is shown.