Pressure Stabilization Method for Steady Viscous Flows in a System of Pipes

This paper deals with a singular perturbation of the stationary Stokes and Navier-Stokes systems. The term ε²Δp is added to the continuity equation, where ε is a small parameter. For a domain with cylindrical outlets to infinity and exponentially decaying data, existence and uniqueness of solutions under flux conditions at infinity are established for the linear problem and also for the nonlinear problem in the case of small data. Asymptotically exact estimates are proved for ε tending to zero. For sufficiently regular data, these estimates imply the convergence in H _loc ^5/2−δ for the velocity parts and in H _loc ^3/2−δ for the pressure parts, respectively. Bibliography: 17 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 107–133.     

An Efficient Numerical Model for Planar Spiral Inductor On-Chip

In this paper, a very useful numerical technique has been developed for analyzing the transient characteristics of a planar-spiral inductor on-chip. A locally conformal technique and an alternating-direction implicit scheme are applied to the finite-difference time-domain method. A formulation for solving three dimensional Maxwell’s equations is proposed. Using the proposed method, various parameters of the planar-spiral inductors have been analyzed and an equivalent circuit, which includes frequency-independent circuit elements, has been introduced. Highly computational efficiency is implemented. Numerical results show excellent agreement with the measured data over a wide frequency range.     

Variational Derivative for Buckling Loads of Beams and Frames

The finite-difference method is a numerical technique for obtaining approximate solutions to differential equations. The main objective of the present study is to give a new aspect to the finite-difference method by using a variational derivative. By applying this formulation, accurate values of the buckling loads of beams and frames with various end supports are obtained. The performance of this formulation is verified by comparison with numerical examples in the literature __________ Published in Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 139–144, July 2005.     

Monotone Nonincreasing Random Fields on Partially Ordered Sets. I

For an arbitrary poset H and measure ρ on H × _R (where _R is the real axis), we construct a monotone decreasing stochastic field η_ρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field η_ρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 92–143.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.