On inf-sup conditions and projections in the theory of mixed finite-element methods

We indicate constraints on the space of finite elements providing the validity of discrete inf-sup conditions and the existence of projections specific for mixed finite-element methods. We consider both conformal and nonconformal approximations. We suggest a definition of special projections onto the vector space of finite elements which provides their existence under quite general conditions without determining the degrees of freedom of the elements. © 2014 Pleiades Publishing, Ltd

Symmetries and cycles of the renormalization group in a fermionic hierarchical model

We study singular points and symmetries of the renormalization group mapping in a fermionic hierarchical model. This mapping taken at the renormalization group singular point and with the renormalization group parameter α = 1 generates cycles of arbitrary lengths

Sharp estimates for the polynomial approximation in weighted Sobolev spaces

© 2015, Pleiades Publishing, Ltd. We obtain sharp estimates for the accuracy of the best approximation of functions by algebraic polynomials on an interval, the half-line, and the entire line in weighted Sobolev spaces with Jacobi, Laguerre, and Hermite weights, respectively. We show that the orthogonal polynomials associated with these weights form orthogonal bases in the respective weighted Sobolev spaces. We obtain sharp estimates of Markov–Bernstein type

HDG schemes for stationary convection-diffusion problems

© Published under licence by IOP Publishing Ltd.For stationary linear convection-diffusion problems, we construct and study a hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability and accuracy of the schemes

Discontinuous mixed penalty-free Galerkin method for second-order quasilinear elliptic equations

Discrete schemes for finding an approximate solution of the Dirichlet problem for a second-order quasilinear elliptic equation in conservative form are investigated. The schemes are based on the discontinuous Galerkin method (DG schemes) in a mixed formulation and do not involve internal penalty parameters. Error estimates typical of DG schemes with internal penalty are obtained. A new result in the analysis of the schemes is that they are proved to satisfy the Ladyzhenskaya-Babuska-Brezzi condition (inf-sup) condition. © 2013 Pleiades Publishing, Ltd

On the boundary conditions for Navier-Stokes equations in stream function-vorticity variables in simulation of a flow around a system of bodies

A method of determining the boundary conditions for the Navier-Stokes equations in stream function-vorticity variables, used for simulation of a nonstationary, asymmetric laminar flow of an incompressible viscous fluid around bodies, has been proposed. Universal relations for desired functions on surfaces around which the stream flows, independent of the method of spatial discretization, have been obtained. © 2005 Springer Science+Business Media, Inc

Solution of the vector eigenmode problem for cylindrical dielectric waveguides based on a nonlocal boundary a condition

The original problem in an unbounded domain is reduced to a problem in a disk convenient for numerical solution. The problem thus obtained is a parametric eigenvalue problem with a nonlocal boundary condition nonlinear in the spectral parameter. The analysis of the problem is based on the spectral theory of compact self-adjoint operators. The existence of a spectrum of the problem is proven, and the properties of the dispersion curves are studied. Copyright © 2002 by MAIK "Nauka/Inrerperiodica"

Finite element approximation and iterative method solution of elliptic control problem with constraints to gradient of state

© 2015, Pleiades Publishing, Ltd. An optimal control problem with distributed control in the right-hand side of Poisson equation is considered. Pointwise constraints on the gradient of state and control are imposed in this problem. The convergence of finite element approximation for this problem is proved. Discrete saddle point problem is constructed and preconditioned Uzawa-type iterative algorithm for its solution is investigated

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

© 2018 Walter de Gruyter GmbH, Berlin/Boston. The original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically