Band 3, Heft 4

A parameterized Newton’s method to guarantee convergence to an eigenpair of a real symmetric matrix in a designated interval has been developed. The method is parametric in nature and with appropriate choices of parameters, the classical methods such as Newton’s method and the Rayleigh quotient method can easily be recovered.

A DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp discretization, obtain a fast preconditioner-solver for faces by Kinterpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently.

The application of the numerical functional integration method to the solution of differential equations in quantum physics is discussed. We have developed a method of numerical evaluation of functional integrals in abstract complete separable metric spaces, which proves to have important advantages over the conventional Monte Carlo method of path integration. One of the considered applications is the investigation of open quantum systems (OQS), i.e., systems interacting with their environment. The density operator of OQS satisfies the known Lindblad differential equation. We have obtained the expression for matrix elements of this operator in the form of the double conditional Wiener integral and considered its application to some problems of nuclear physics. Another application is the solution of the Scr¨odinger equation with imaginary time and anticommuting variables for studying many-fermion systems. We have developed a numerical method based on functional integration over ordered subspaces. The binding energies of some nuclei are computed using this method. Comparison of the results with those obtained by other authors and with experimental values is presented.

For difference schemes the initial-boundary problem for quasi-linear parabolic-type equations, ’a priori weight estimates’ of the error have been found. These estimates show how much the accuracy of difference schemes near the boundary of a time rectangle is higher than in the middle of it. Sufficient conditions of smoothness of the coefficients and the right-hand side of the quasi-linear parabolic equation and the initial conditions have been found. These conditions ensure a correctness of these a priori estimates.

We consider the hp-version of the discontinuous Galerkin finite element approximation of boundary value problems for the biharmonic equation. Our main concern is the a priori error analysis of the method, based on a nonsymmetric bilinear form with interior discontinuity penalization terms. We establish an a priori error bound for the method which is of optimal order with respect to the mesh size h , and nearly optimal with respect to the degree p of the polynomial approximation. For analytic solutions, the method exhibits an exponential rate of convergence under p- refinement. These results are shown in the DG-norm for a general shape regular family of partitions consisting of d-dimensional parallelepipeds. The theoretical results are confirmed by numerical experiments. The method has also been tested on several practical problems of thin-plate-bending theory and has been shown to be competitive in accuracy with existing algorithms.

In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is remarkably well-conditioned, allowing one to combine the superior least-squares performance of Chebyshev polynomials with the geometrical insight of the Bernstein form. We also compare it to other basis transformations such as Bernstein-Hermite, power-Hermite, and Bernstein–Legendre basis transformations.

High-order accurate methods for convection-dominated flow problems have the potential to reduce the computational effort required for a given order of solution accuracy. The state of the art in this field is more advanced for Eulerian methods than for semi-Lagrangian methods. In this paper, we introduce a new higher order approach that is based on combining the modified method of characteristics with the gradients computed during solution reconstruction. Special attention is given to the class of third-order schemes which uses a higher-order method to calculate the departure points and a high-order polynomial reconstruction to interpolate the solution at the characteristic feet. Numerical experiments are carried out on several benchmark transport-diffusion equations including incompressible Navier-Stokes problems. The computed results support our expectations for a robust and high-order accurate method.

A new product triangular iterative methods for solving nonsymmetric linear systems of equations with positive real coefficient matrices based on the skewsymmetric part of the initial matrix are proposed. The convergence analysis for the new methods is presented.