Band 6, Heft 4

In this paper a simple equilibrium finite element on quadrilaterals is introduced and analysed. An optimal a priori error estimate has been obtained for arbitrary regular quadrilaterals.

Tomography is a classic inverse problem in which multiple density projections of an object are processed to infer some approximation of the original. We consider the highly sparse inverse problem of single angle projection, but seek to reduce the ambiguity through multiple time observations in a dynamic system of known or partially known dynamics. In this work we solve the planar problem by optimization techniques based on a gradient-free multi-directional search algorithm to minimize our nonlinear functional. We demonstrate convincingly successful numerical examples to support our relatively simple technique.

A spectral boundary problem on the eigenfield of an inflated/deflated stretched circular membrane, which is clamped to a circular cylindrical cavity filled with a liquid, is examined. The paper presents an operator formulation of the problem and proposes a new semi-analytical approximate method. The method captures singular behavior of the solution in the pole and at the fastening contour of the membrane.

We develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

Asymptotic stability of linear three-level operator-difference schemes is investigated in the case of commutative operators. Some new a priori estimates are obtained.

This paper presents a new robust multigrid technique for solving boundary value problems in a black box manner. To overcome the problem of robustness, the technique is based on the incorporation of adaptation of boundary value problems to numerical methods, control volume discretization and a new multigrid solver into a united computational algorithm. The special multiple coarse grid correction strategy makes it possible to obtain problem-independent transfer operators. As a result, most modes are approximated on coarse grids to make the task of the smoother on the finest grid the least demanding. A detailed description of the robust multigrid technique and examples of its application for solving benchmark problems are given in the paper.

In this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.