Left-definite boundary conditions of Sturm-Liouville problems when the interval shrinks to a point

On the condition that the interval of the problem shrinks to a point, we investigated the separated boundary conditions S α,β of left-definite Sturm-Liouville problem, and answered the following question: Is there a c 0 ∈ J such that S α,β is always left-definite or semi-left-definite for the Sturm-Liouville equation for each c ∈ (a, c 0 )?     

Finite-difference approximation of dirichlet observation problems for weak solutions to the wave equation subject to Robin boundary conditions

For the wave equation with variable coefficients subject to Neumann and Robin boundary conditions, two mutually dual problems are considered: the Dirichlet observation problem with weak generalized solutions and the control problem with strong generalized solutions. Both problems are approximated by finite differences preserving the duality relation. The convergence of the approximate solutions is established in the norms of the corresponding dual spaces.     

On an approach to the study of the solvability of boundary value problems for the system of differential equations of the 3D theory of elasticity

We study the solvability of a boundary value problem for a system of second-order linear partial differential equations. A theorem on the existence of a solution of the problem is proved. The method used in the study is to reduce the original system of equations to a system of 3D singular integral equations, whose solvability can be proved with the use of the notion of symbol of a singular operator.     

Parallel linear multigrid by agglomeration for the acceleration of 3D compressible flow calculations on unstructured meshes

In this paper, we report on our recent efforts concerning the design of parallel linear multigrid algorithms for the acceleration of 3-dimensional compressible flow calculations. The multigrid strategy adopted in this study relies on a volume agglomeration principle for the construction of the coarse grids starting from a fine discretization of the computational domain. In the past, this strategy has mainly been studied in the 2-dimensional case for the solution of the Euler equations (see Lallemand et al. [6]), the laminar Navier–Stokes equations (see Mavriplis and Venkatakrishnan [12]) and the turbulent Navier–Stokes equations (see Carré [1], Mavriplis [10] and Francescatto and Dervieux [4]). A first extension to the 3-dimensional case is presented by Mavriplis and Venkatakrishnan in [13] and more recently in Mavriplis and Pirzadeh [11]. The main contribution of the present work is twofold: on the one hand, we demonstrate the successful extension and application of the multigrid by a volume agglomeration principle to the acceleration of complex 3-dimensional flow calculations on unstructured tetrahedral meshes and, on the other hand, we enhance further the efficiency of the methodology through its adaptation to parallel architectures. Moreover, a nontrivial aspect of this work is that the corresponding software developments are taking place in an existing industrial flow solver. This revised version was published online in June 2006 with corrections to the Cover Date.