G. M. Kobelkov

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term

Relaxation folding and the principle of the minimum rate of energy dissipation for conformational motions in a viscous medium

\alpha\boldmath{u}

Formation of collective conformational degrees of freedom during macromolecular chain folding dynamics in a viscous medium

in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent of

Numerical Solution of a Tidal Wave Problem

\alpha

Implementation of a least-squares finite element method for solving the Stokes problem with a parameter

. Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are provided.

On modifications of the Navier–Stokes equations

A numerical simulation of the folding of a model polymer chain of 50 units with valence bonds of a fixed length and fixed valence angle values has been performed using the strong friction approximation. The rate of energy dissipation in the system has been analyzed for conformational motions along a trajectory determined by the equations of mechanics and the trajectories characterized by random and variable deviations from the mechanical path. The validity of the principle of the minimum average rate of the energy dissipation for the conformational relaxation of a macromolecule in a viscous medium has been demonstrated. A profile of the relaxation energy funnel for the folding of a macromolecular chain has been constructed. Slow and rapid stages of folding could be distinguished in the energy funnel profile; the final state was separated from the nearest conformations of the folded chain by an energy gap.

Finite difference approximation of tidal wave equations on unstructured grid in spherical coordinates

Computational methods were used to study the dynamics of the formation of the collective conformational degrees of freedom in the relaxation folding of a model biopolymer chain of 50 nodes in a viscous medium; the model has been described previously. Collective conformational motions of the nodes were shown to arise due to friction forces in a viscous medium. The collective motions have several typical forms, including a wave of differently directed motions of chain nodes that propagates from one end of the chain to another (like a soliton) in response to a pertubation in terminal group position. Individual nodes located at the middle of the chain make approximately equal contributions to the total energy dissipation rate. The end nodes contribute approximately 2–4 times more than internal nodes to the total energy dissipation. The results of numerical experiments are consistent with the theoretical concept developed earlier to describe the dynamics of linear macromolecular chains in a viscous medium in the limit of a very large number of nodes.

Efficient methods for solving elasticity theory equations

Publisher Summary This chapter reviews the numerical solution of a tidal wave problem. The traditional model in oceanography describes behavior of ocean circulation in the spherical coordinate system—x, y, z, which is very complicated for direct simulation. So the usual approach for its solution consists of splitting schemes. In this way, the original problem is split into a number of sub problems to be solved step by step. Nevertheless, even these sub problems are rather difficult for numerical solution, so appropriate results may be obtained with the help of powerful computers, in particular, multiprocessor ones. One of the problems arising after splitting consists in taking into account effects induced by tidal waves.

2. Numerical simulation of spacial motion of a thread

The implementation of a least-squares finite element method for solving the generalized stationary Stokes problem (i.e. the Stokes problem with an additional term α u in the motion equation, where α is a big parameter and u is the velocity vector function) is considered. The basis of this method is the reduction of the second-order boundary value problem to a system of first-order partial differential equations and the minimization of the residuals of these equations in some finite element space by the least-squares method. The main advantage of this approach consists in the fact that the same approximating space is used for both the velocity and the pressure. The condition number of the resulting system of linear algebraic equations depends on the big parameter α; an efficient preconditioner for this system is constructed. Copyright

Justification of a splitting scheme for ocean dynamics equations

Abstract A modification of the Navier-Stokes equations being in some sense generalization of the Ladyzhenskaya modification [2] is considered. Namely,we modify the elliptic operator in the first two motion equations in the same way as Ladyzhenskaya, but in two variables (not three), while the third motion equation is not changed. For the mixed boundary value problem in a bounded domain, existence and uniqueness of a solution ‘in the large’ is proved. The case of viscosity coefficient different in vertical and horizontal directions is also considered.