A tensor product generalized ADI method for elliptic problems on cylindrical domains with holes

We consider solving second order linear elliptic partial differential equations together with Dirichlet boundary conditions in three dimensions on cylindrical domains (nonrectangular in x and y) with holes.We approximate the partial differential operators by standard partial difference operators. If the partial differential operator separates into two terms, one depending on x and y, and one depending on z, then the discrete elliptic problem may be written in tensor product form as (T_z⊗I + I⊗A_xy) U=F. We consider a specific implementation which uses a Method of Planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction. We establish the convergence of the Tensor Product Generalized Alternating Direction Implicit iterative method applied to such discrete problems. We show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems.     

Optimum parameters for the generalized ADI method

Summary. In the context of the generalized ADI method, we are concerned with the problem of finding in the set of rational functions r with numerator degree m and denominator degree n an element that minimizes where E,F are disjoint real intervals. By extending a recent analysis by Levin and Saff, we present an explicit formula for choosing the pair (m,n) for given m +n. Furthermore, we provide a characterization of and a Remes type algorithm for its determination. Extensive numerical computations furnish some comparison of with asymptotically optimal solutions based on Fejér-Walsh and Leja-Bagby points. Received September 6, 1996 / revised version received June 30, 1997     

Tensor product methods for stochastic problems

In the solution of stochastic partial differential equations (SPDEs) the generally already large dimension N of the algebraic system resulting from the spatial part of the problem is blown up by the huge number of degrees of freedom P coming from the stochastic part. The number of degrees of freedom of the full system will be NP, which poses severe demands on memory and processor time. We present a method how to approximate the system by a data-sparse tensor product (based on the Karhunen-Loève decomposition with M terms), which uses only memory in the order of M (N + P), and how to keep this representation also inside the iterative solvers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the ADI method for Sylvester equations

This paper is concerned with the numerical solution of large scale Sylvester equations AX−XB=C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) [22] and Li and White (2002) [20] demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz’s shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions.     

A fourth order ADI method for semidiscrete parabolic equations

A fourth order fourstep ADI method is described for solving the systems of ordinary differential equations which are obtained when a (nonlinear) parabolic initial-boundary value problem in two dimensions is semi-discretized. The local time-discretization error and the stability conditions are derived. By numerical experiments it is demonstrated that the (asymptotic) fourth order behaviour does not degenerate if the time step increases to relatively large values. Also a comparison is made with the classical ADI method of Peaceman and Rachford showing the superiority of the fourth order method in the higher accuracy region, particularly in nonlinear problems.     

Tensor product of partially-additive monoids

Partially-additive monoids (pams) were introduced by Arbib and Manes in order to provide an algebraic semantics for programming languages. In this paper, we prove that the categoryP a m of pams and additive maps is a closed category whose monoids are partially-additive semirings. We follow the tensor product construction of R. Guitart [7] for categories of algebras which generalize the case of modules. Nevertheless, the problem here is more difficult owing to the fact that pams are partial algebras rather than algebras. Thus, we have to make some modifications to Guitart’s approach.     

Tensor product of C-injective modules

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness     

Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

High-order ADI method for stream-function vorticity equations

Many recent works have demonstrated the efficiency of high-order compact (HOC) difference schemes on the stream-function and vorticity formulation of 2-D incompressible Navier-Stokes equations. HOC discretizations induce cross spatial derivatives which are treated explicitly in most ADI schemes. Recently, Karaa and Zhang proposed a fourth-order ADImethod for solving convection-diffusion problems efficiently. In this work, we extend this method to the solution of incompressible Navier-Stokes equations. The driven flow in a square cavity is used as a model problem and numerical results are compared with other results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tensor product Gauss-Lobatto points are Fekete points for the cube

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the -dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejér in 1932, but the extension to higher dimensions in non-trivial.