### Book chapter

## Применения топологии в одной численной схеме решения задач механики сплошных сред

Finite element numerical schemes for solving the continuum mechanics problems are discussed. One of the authors developed a method of acceleration of calculations which uses the simplicial mesh inscribed in the original cubic cell partitioning of a three-dimensional body. In this work it is shown that the obstacle to the construction of this design may be described in terms of modulo 2 homology groups. The method of removing the obstacle is proposed.

### In book

The volume contains articles of scientific staff and faculty of the Department of Computer Science and Applied Mathematics and Scientific-Educational Center of computer modeling of unique buildings and complexes of Moscow State University of Civil Engineering (National Research University), devoted to actual problems of applied mathematics and computational mechanics.

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large x. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.

The study is carried out by the first author within The National Research University Higher School of Economics' Academic Fund Program in 2012-2013, research grant No. 11-01-0051.

We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded $n$-dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error estimates $O(\tau^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large x. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials. The study is carried out by the first author within The National Research University Higher School of Economics' Academic Fund Program in 2012-2013, research grant No. 11-01-0051.