Volume 4, Issue 2

In this paper we consider numerical algorithms for solving the system of nonlinear PDEs, arising in modeling of liquid polymer injection. We investigate the particular case where a porous preform is located within the mould, so that the liquid polymer is flowing through a porous medium during the filling stage. The nonlinearity of the governing system of PDEs is due to the non-Newtonian behavior of the polymer, as well as to the moving free boundary. The latter is related to the penetration front, and a Stefan type problem is formulated to take into account. A finite-volume method is used to approximate the given differential problem. Results from numerical experiments are presented. We also solve an inverse problem and present algorithms for determination of the absolute preform permeability coefficient for the case where the velocity of the penetration front is known from the measurements. In both considered cases (direct and inverse problems), we focus on the specificity related to the non-Newtonian behavior of the polymer. For completeness, we also discuss the Newtonian case. Results of some experimental measurements are presented and discussed.

For two new effcient methods for solving initial value problems in a Hilbert or Banach spaces based on a Sinc quadrature for an improper Dunford-Cauchy integrals over a path enveloping the spectrum of the operator we give a new unified estimate in the case of a Hilbert space.

A mixed hybrid finite element method of the lowest order is studied for the Signorini problem. An iterative method with a preconditioner being a classical finite element approximation of the Laplace operator is constructed. A multistage iterative procedure for the mixed hybrid finite element scheme is constructed, the rate of convergence and the complexity of this method are analysed.

The stability of linear two- and three-level operator difference schemes in a Hilbert space has been investigated. A few a priori estimates of the global and asymptotic stability in various energy norms have been obtained.

In this paper we derive the matrix of transformation of the Jacobi polynomial basis form into the Bernstein polynomial basis of the same degree n and vice versa. This enables us to combine the superior least-squares performance of the Jacobi polynomials with the geometrical insight of the Bernstein form. Application to the inversion of the Bézier curves is given.

The flow of a steady, incompressible, viscous, electrically conducting fluid past a sphere in the presence of uniform magnetic field parallel to the undisturbed flow is investigated using the finite difference method. The multigrid method with a defect correction (DC) technique is used to achieve the second order accurate solution. The Hartmann number M is used as the perturbation parameter. It has been found that the increase of magnetic field decreases the wake length (L) and increases the drag coefficient. The graphs of streamlines, vorticity lines, drag coefficient, wake length, surface pressure and surface vorticity are presented and discussed

We develop a discrete analog of the differential calculus and use this to develop arbitrarily high-order approximations to Sturm–Liouville boundary-value problems with general mixed boundary conditions. An important feature of the method is that we obtain a discrete exact analog of the energy inequality for the continuum boundary-value problem. As a consequence, the discrete and continuum problems have exactly the same solvability conditions. We call such discretizations mimetic. Numerical test confirm the accuracy of the discretization. We prove the solvability and convergence for the discrete boundary-value problem modulo the invertibility of a matrix that appears in the discretization being positive definite. Numerical experiments indicate that the spectrum of this matrix is real, greater than one, and bounded above by a number smaller than three.