Abstract: In time-domain goal-oriented adaptivity, it is essential to represent the error in the quantity of interest as an integral over the whole space-time domain. In that way, we can express the global error as a sum of local element contributions, and perform adaptivity. A space-time variational formulation provides such integral representation. Many authors employ implicit methods in time for performing goal-oriented adaptivity, like Backward Euler or Crank-Nicholson, as it is well known that these methods have variational structure. The Galerkin formulation of explicit methods in time for partial differential equations, however, remains elusive. In this work, we first construct a Petrov-Galerkin formulation for parabolic problems that is equivalent to the Forward Euler method in time (first order Runge-Kutta). Then, we derive an error representation and an explicit goal-oriented adaptive algorithm, enabling dynamic meshes in space. Some numerical results are provided for the 1D advection-diffusion equation to illustrate the proposed explicit algorithm. Finally, we provide an overview of how to build other Runge-Kutta methods using a variational formulation and follow a similar goal-oriented procedure.

Abstract: We focus on three-dimensional isogeometric analysis with tensor product C^k B-spline basis functions. We solve the computational problem with multi-frontal direct solver using the ordering obtained from the element partition trees. The trees have particular elements at the leaves, the entire mesh at the roof, and recursive partitions at internal nodes. The ordering obtained by browsing the tree in post-order, from leaves up to the root, results in a lower computational cost of the LU factorization than state-of-the art orderings obtained from the spare matrix analysis. We propose the algorithm that plots the map of FLOPS per mesh nodes using the trees, which in turn allows to identify the computationally expensive mesh nodes. Finally, we modify the algorithm of refined isogeometric analysis to introduce C0 separators at expensive mesh nodes, which reduces the cost of the solver by order of magnitude, while maintaining the numerical accuracy.

Abstract: To describe the diversity of opinions and dynamics of their changes in a society, there exist different approaches — from macroscopic laws of political processes to individual–based cognition and perception models. In this paper, we propose mesoscopic individual–based model of opinion dynamics which tackles the role of context by considering influence of different sources of information during life cycle of agents. The model combines several sub–models such as model of gen-eration and broadcasting of messages by mass media, model of daily activity, contact model based on multiplex network and model of information processing. To show the applicability of the approach, we present two scenarios illustrating the effect of the conflicting strategies of informational influence on a population and polarization of opinions about topical subject.

Abstract: In the paper, heuristic algorithm for tensor product approximation with B-spline basis functions of three-dimensional material data is presented. The algorithm has an application as a preconditioner for implicit dynamics simulations of a non-linear flow in heterogeneous media using alternating directions method. As the simulation use case non-stationary problem of liquid fossil fuels exploration with hydraulic fracturing is considered. Presented algorithm allows to approximate the permeability coefficient function as a tensor product what in turn allows for implicit simulations of the Laplasjan term in the partial differential equation. In the consequence the number of time steps of the non-stationary problem can be reduced, while the numerical accuracy is preserved.