Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications

In this paper, we consider fully discrete approximations of abstract evolution equations, by means of a quasi non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Hence, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We set the problem in the very general and abstract setting of pseudo-monotone operators, which allows for a unified treatment of several evolution problems. The examples — which fit into our setting and which motivated our research — are problems describing the motion of incompressible fluids, since the quasi non-conforming approximation allows to handle problems with prescribed divergence. Our abstract results for pseudo-monotone operators allow to show convergence just by verifying a few natural assumptions on the operator time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be easily performed. The results of some numerical experiments are reported in the final section.

WORDING OF AXIOLOGICAL DISTANCING IN MEDIA TEXTS ON POLITICALLY SENSITIVE TOPICS

The current state of linguistic pragmatics is at the stage of active development of the terminological basis and theoretical and methodological foundations. One of the notions used in the generally accepted methodology within this area is "axiological distancing". The use of this notion and the corresponding method of analysis allows us to fully disclose the strategies and tactics of constructing texts to influence the reader, analyse the manipulative strategies of the author, assess its objectivity and impact, etc. The results of the analysis of the selected texts allow us to make the conclusion that the ideological opponent in the texts under analysis is the abstract notion of a religious (Muslim) fanatic-terrorist in the eyes of a Christian and a European. It finds a concrete linguistic embodiment described and classified in the article. Having fixed the criteria of axiological classification of the opponent and the main value opposition, we passed to the analysis of the wording of the resistance to the opponent as an aspect of axiological distancing. The analysis of axiological distancing proved that the identification of the opponent's features by studying the semantics of lexical units used in the reference reveals a possible communicative effect and manipulative influence on the reader. According to the identified opposition pattern, the authors form a linguistic expression of the violation of the distance between the opponents, i.e. their physical influence and spatial approximation (we made the classification). The article describes the steps of building the axiological opposition "we" – "they", which is represented by such elements as "West" – "East", "Christianity" – "Islam", "freedom" – "dependence". Special attention is paid to the ways of providing objective reflection of events and keeping to the standards of journalistic ethics.

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.

A class of parabolic evolutionary quasivariational inequalities in contact mechanics

In this paper, we obtain an existence and uniqueness of the solution for a class of parabolic evolutionary quasivariational inequalities in contact mechanics under some mild conditions. We also study an error estimate for the parabolic evolutionary quasivariational inequality by employing the forward Euler difference scheme and the element-free Galerkin spatial approximation.

Calculation of heat transfer in nanoscale heterostructures

The article discusses the calculation of the temperature regime in nanoscale AlAs/GaAs binary heterostructures. When modeling heat transfer in nanocomposites, it is important to take into account that heat dissipation in multilayer structures with layer sizes of the order of the mean free path of energy carriers (phonons and electrons) occurs not at the lattice, but at the layer boundaries (interfaces). In this regard, the use of classical numerical models based on the Fourier law is limited, because it gives significant errors. To obtain more accurate results, we used a model in which the heat distribution was assumed to be constant inside the layer, while the temperature was stepwise changed at the interfaces of the layers. A hybrid approach was used for the calculation: a finite−difference method with an implicit scheme for time approximation and a mesh−free model based on a set of radial basis functions for spatial approximation. The calculation of the parameters of the bases was carried out through the solution of the systems of linear algebraic equations. In this case, only weights of neuroelements were selected, and the centers and «widths» were fixed. As an approximator, a set of frequently used basic functions was considered. To increase the speed of calculations, the algorithm was parallelized. Calculation times were measured to estimate the performance gains using the parallel implementation of the method.

Analysis of a Poro-Thermo-Viscoelastic Model of Type III

In this work, we numerically study a thermo-mechanical problem arising in poro-viscoelasticity with the type III thermal law. The thermomechanical model leads to a linear system of three coupled hyperbolic partial differential equations, and its weak formulation as three coupled parabolic linear variational equations. Then, using the finite element method and the implicit Euler scheme, for the spatial approximation and the discretization of the time derivatives, respectively, a fully discrete algorithm is introduced. A priori error estimates are proved, and the linear convergence is obtained under some suitable regularity conditions. Finally, some numerical results, involving one- and two-dimensional examples, are described, showing the accuracy of the algorithm and the dependence of the solution with respect to some constitutive parameters.

Stochastic and deterministic kinetic energy backscatter parameterizations for simulation of the two-dimensional turbulence

The problem of modelling 2D isotropic turbulence in a periodic rectangular domain excited by the forcing pattern of prescribed spatial scale is considered. This setting could be viewed as the simplest analogue of the large scale quasi-2D circulation of the ocean and the atmosphere. Since the direct numerical simulation (DNS) of this problem is usually not possible due to the high computational costs we explore several possibilities to construct coarse approximation models and corresponding subgrid closures (deterministic or stochastic). The necessity of subgrid closures is especially important when the forcing scale is close to the cutoff scale of the coarse model that leads to the significant weakening of the inverse energy cascade and large scale component of the system dynamics. The construction of closures is based on the a priori analysis of the DNS solution and takes into account the form of a spatial approximation scheme used in a particular coarse model. We show that the statistics of a coarse model could be significantly improved provided a proper combination of deterministic and stochastic closures is used. As a result we are able to restore the shape of the energy spectra of the model. In addition the lagged auto correlations of the model solution as well as its sensitivity to external perturbations fit the characteristics of the DNS model much better.