A method for the determination of the minimax

Translated from Issledovaniya po Prikladnoi Matematike, No. 7, pp. 19–23, 1979.     

A penalty method for grid matching in the mixed Herrmann-Miyoshi scheme

A penalty method for mixed finite element methods is formulated and studied. The Herrmann-Miyoshi scheme for the biharmonic equation is considered. The main idea is to build a perturbed problem with two parameters playing the role of penalties. The perturbed problem is constructed by replacing principal conditions in the mixed variational formulation at the interface by natural conditions containing parameters. Discretization of the perturbed problem is effected by a finite element method. Estimates for the norm of the difference between the solutions of a discrete perturbed problem and of an initial value problem are derived depending on the mesh size and penalties. Recommendations are given as to how to choose penalties so as to fit a mesh size.     

A quadrature method for solving integral equations of the mixed type

This paper is dedicated to quadrature methods for solving integro-operational equations. We construct computational schemes of quadrature methods and substantiate them theoretically in the sense of the general theory of approximate methods developed by L. V. Kantorovich and B. G. Gabdulkhaev.     

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the varia-tional inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.     

A particular solution of the relativistic theory of gravity obtained by the indefinite coordinate method

A particular solution of the equations of the relativistic theory of gravity describing the gravitational field of an infinitely thin source arranged along the z-axis is obtained by the indefinite coordinate method. In this field, the motion of massive particles is studied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 3, pp. 501–512, September, 1997.     

A refined mixed finite element method for the boussinesq equations in polygonal domains

This paper deals with the mixed formulation of the Boussinesqequations in two-dimensional polygonal domains and its numericalapproximation. The steady solution has a singular behaviournear the corner points so that we show that it belongs to appropriateweighted Sobolev spaces. Since uniform meshes lead to a slowconvergence rate, we derive appropriate refinement rules onthe meshes near the corner points in order to restore the quasi-optimalrate of convergence. A numerical test is finally presented whichconfirms the theoretical convergence rates. Received 26 November 1999. Accepted 25 August 2000.     

A third-order mixed finite-element method for the numerical solution of the biharmonic problem

This paper is devoted to the introduction of a mixed finite element for the solution of the biharmonic problem. We prove optimal rate of convergence for the element. The mixed approach allows the simultaneous approximation of both displacement and tensor of its second derivatives.     

A new method for estimating the forecast quality with consideration for the errors of calculating the unknown parameters

In this study, the problem of estimating the forecast accuracy of a model is considered. A widespread practice is to approximate the population expectation of the forecast accuracy by the sample expectation, which is equivalent to the uniform consideration for the deviations of the forecast from the exact value of a quantity for all time moments. If the vector of unknown parameters is estimated at each step only from the preceding observations, the significance of the deviations is not the same at all time moments. In this study, we propose a method that takes into account the forecast errors with different weights. The problem of constructing the most accurate estimate of the forecast quality, a parameter from which the condition for the optimal weights can be derived, is formalized. Monte-Carlo experiments are used to compare the accuracy of the methods for estimating the forecast quality in the cases when the observations are taken into account with the same weights, with optimum weights, and with the weights calculated using a numerical procedure.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

A posteriori error estimation for the dual mixed finite element method of the Stokes problem

This Note presents an a posteriori error estimator of residual type for the stationary Stokes problem using the dual mixed FEM. We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution. To cite this article: M. Farhloul et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A characteristic mixed element method for displacement problems of compressible flow in porous media

A new characteristic mixed element scheme is formulated to solve numerically displacement problems of compressible fluids in porous media. A new mixed finite element method is introduced to solve the pressure equation of parabolic type, in which the mixed element system is symmetric positive definite and the pressure equation is separated from the flux equation. The modified method of characteristics is used to treat convection-dominated diffusion equations of the concentrations. The convergence with optimal accuracy is proved under the general condition. Project supported in part by China State Major Key Project for Basic Researches, Doctoral Station Foundation and TCTPF of China State Education Commission.     

A method for the experimental determination of the viscoelastic properties of a cylindrical sample

Forced oscillations of a cylinder, clamped to a vibrating base and loaded on top with a certain mass, is considered using a two-dimensional deformation model. The boundary conditions are specified over the whole cylinder surface, and the equation of the balance of forces on its outer surface is used. An algorithm for solving the problem using the collocation method is proposed. It is shown that the accuracy with which the modulus of elasticity and the loss factor are determined is much higher than when using methods based on the one-dimensional model, since the proposed method does not require a knowledge of the dynamic form factor.