Том 59, № 1 (2019)

A novel method for deriving a posteriori error bounds for approximate solutions of reaction–diffusion equations is proposed. As a model problem, the problem \( - \Delta u + \sigma u = f\) in \(\Omega \), \({{\left. u \right|}_{{\partial \Omega }}} = 0\) with an arbitrary constant reaction coefficient \(\sigma \geqslant 0\) is studied. For the solutions obtained by the finite element method, bounds, which are called consistent for brevity, are proved. The order of accuracy of these bounds is the same as the order of accuracy of unimprovable a priori bounds. The consistency also assumes that the order of accuracy of such bounds is ensured by test fluxes that satisfy only the corresponding approximation requirements but are not required to satisfy the balance equations. The range of practical applicability of consistent a posteriori error bounds is very wide because the test fluxes appearing in these bounds can be calculated using numerous flux recovery procedures that were intensively developed for error indicators of the residual method. Such recovery procedures often ensure not only the standard approximation orders but also the superconvergency of the recovered fluxes. The advantages of the proposed family of a posteriori bounds are their guaranteed sharpness, no need for satisfying the balance equations in flux recovery procedures, and a much wider range of efficient applicability compared with other a posteriori bounds.

The gradient projection method is generalized to the case of nonconvex sets of constraints representing the set-theoretic intersection of a smooth surface with a convex closed set. Necessary optimality conditions are studied, and the convergence of the method is analyzed.

An inverse problem for Laplace’s equation in a doubly connected two-dimensional domain is considered. Given Dirichlet and Neumann data specified on the known outer boundary of the domain, the task is to determine an unknown inner boundary on which the function takes a constant value. The uniqueness of the solution to this inverse problem is proved. An iterative numerical method for determining the unknown boundary is proposed. Numerical results are presented.

Model equations describing convective transport are used to analyze the approximation errors of an explicit numerical scheme and various implicit schemes with the same approximation of spatial derivatives. It is shown that, under time step constraints determined by the Courant–Friedrichs–Lewy condition, the implicit scheme is inferior in accuracy to the explicit one and, with a further increase in the time step, the accuracy of simulated convective processes degrades substantially. Two methods for implementing the marching procedure in time are considered, namely, a fractional step in the case of an explicit scheme and a dual step in the case of an implicit scheme. It is shown that the fractional step method is efficient only on grids with a scatter of cell sizes of 100–1000. For the numerical solution of problems with no-slip conditions on solid walls (scatter of cell sizes of 10⁴–10⁵), two approaches are proposed: an implicit scheme with a dual step in all cells and an zonal approach, in which a dual step is used in a thin near-wall domain (about 3% of the thickness of a developed turbulent boundary layer), while a fractional step is applied in the rest of the domain. These two approaches are used to compute the flow over an airfoil with flaps. Numerical and experimental data are compared. The accuracy of the numerical results is estimated. The causes of error formation are examined. The domain of efficient application is determined for each of the indicated approaches.

The discrete source method is used to study the influence exerted by nonlocal screening on the optical properties of a linear cluster of nonspherical plasmonic nanoparticles separated by a subnanometer gap. It is shown that deformations of the particles and a reduction in the interparticle gap size lead to an enhanced nonlocal screening effect. It is found that an increase in both scattered and near field intensities is blocked by the nonlocal effect, and deformations of the particles can be used as an alternative to field intensity enhancement.

A method for the quantitative comparison of the spatial geometric structure of two molecules is proposed. It is based on the minimization of a comparison function using the rotation of molecules when their centers of mass are brought into coincidence. The minimizing angles are found using the Rosenbrock method.

The numerical solution of initial value problems is used to obtain compacton and kovaton solutions of K(f ^m, g ⁿ) equations generalizing the Korteweg–de Vries K(u², u¹) and Rosenau–Hyman K(u ^m, u ⁿ) equations to more general dependences of the nonlinear and dispersion terms on the solution u. The functions f(u) and g(u) determining their form can be linear or can have the form of a smoothed step. It is shown that peakocompacton and peakosoliton solutions exist depending on the form of the nonlinearity and dispersion. They represent transient forms combining the properties of solitons, compactons, and peakons. It is shown that these solutions can exist against an inhomogeneous and nonstationary background.