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In this paper, we propose a new mixed finite element (FE) method for diffusion equations on a sequence of two hierarchical polyhedral meshes: a coarse and a fine one. The method is motivated by applications of diffusion equations in basin modelling when the computational domain is a union of ‘horizontal’ layered subdomains and the layers may degenerate. We consider in detail only the case of meshes with distorted and degenerated ‘vertical’ prismatic mesh cells. The discretization on the coarse mesh is derived from the macro-hybrid mixed FE method on the fine mesh with a new definition of the underlying FE spaces. The new mixed FE spaces are subspaces of classical spaces and they are designed by using special homogenization/coarsening algorithms. The accuracy and efficiency of the method is illustrated by numerical results for 2D and 3D diffusion equations.

An initial boundary value problem for a system of nonlinear partial differential equations is formulated in order to study relativistic cylindrical oscillations in plasma. Its approximate solutions are constructed based on a numerical finite difference method and an analytic-numerical perturbation method. It is stated that disruption processes in plasma oscillations and plasma wake waves have a qualitative similarity. Asymptotic estimates for the disruption time of oscillations are obtained from above and from below, which are in accordance with the nonrelativistic case.

An efficient method for modelling incompressible free surface flows is presented. The method unites the projection method for solving the Navier–Stokes equations and the particle level set method for free surface evolution. The method uses adaptively refined hexahedral meshes built on an enhanced octree data structure.

New approximations of Navier–Stokes equations for an incompressible fluid on triangular and tetrahedral grids are proposed using a predictor-corrector method. Estimates of the unconditional stability of constructed schemes and numerical solutions of a two-dimensional problem of fluid flow in a cavity with a moving upper lid are presented. A comparison with results of other authors is given. The problems of monotonicity of grid equations are discussed.

Functional a posteriori estimates have been derived for elliptic and linear parabolic problems in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals, 2000. S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2003. S. Repin, A posteriori error estimates for partial differential equations, 2008. S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, 1997] and some other publications. They provide computable upper bounds of the difference between the exact solution u and any approximation v lying in the admissible (energy) class. This paper is concerned with advanced forms of these estimates, which are discussed within the paradigm of the reaction-diffusion problem. In the first part of the paper, we derive guaranteed and computable upper bounds for problems which admit the decomposition of the domain into a set of simple (e.g., simplicial or polyhedral) subdomains. For this case an upper bound which involves constants in the Poincaré inequality for the corresponding subdomains is obtained. Estimates of this type can be helpful if computations are performed by the domain decomposition method . The second part of the paper is devoted to the derivation of two-sided error bounds in terms of weighted norms . Our analysis shows that the approach based on certain transformations of the basic integral identity earlier developed for energy error norms (see [S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2003. S. Repin, A posteriori error estimates for partial differential equations, 2008]) can be successfully applied to error estimation in weighted norms.