Some new HSOLSSOMs of types hn and 1m u1

In this paper, two new direct construction methods are given for holey self‐orthogonal Latin squares with a symmetric orthogonal mate (HSOLSSOMs). Some new HSOLSSOMs using already known methods are also given. The known existence results for HSOLSSOMs of types 1^m u¹ and hⁿ are improved; for type 1^m u¹ there remain just four possible exceptions with u odd and 3 ≤ u ≤ 15; for type hⁿ, there are just two possible exceptions remaining, for (h, n) = (6, 12) and (6, 18). As a byproduct, the known existence results for three holey mutually orthogonal Latin squares (3 HMOLS) are also improved. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 435–444, 2001     

Single cell discretization of O(kh2 + h4) for the estimates of ${\partial u}\over{\partial n}$ for the two‐space dimensional quasi‐linear parabolic equation

In this article, new stable two‐level explicit difference methods of O(kh² + h⁴) for the estimates of for the two‐space dimensional quasi‐linear parabolic equation are derived, where k > 0 and h > 0 are grid sizes in time and space directions, respectively. We use a single computational cell for the methods, which are applicable to the problems both in cartesian and polar coordinates. The proposed methods have the simplicity in nature and employ the same marching type technique of solution. Numerical results obtained by the proposed methods for several different problems were compared with the exact solutions. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 250–261, 2001     

Efficient linear solvers for incompressible flow simulations using Scott‐Vogelius finite elements

Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, to appear) divergence‐free mixed finite elements may have a significantly smaller discretization error than standard nondivergence‐free mixed finite elements. To judge the overall performance of divergence‐free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((P_k)^d,P _k‐1^disc) Scott‐Vogelius finite element implementations of the incompressible Navier–Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott‐Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor‐Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and \begin{align*}\mathcal{H}\end{align*} ‐LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error Estimates for Finite-Element Navier-Stokes Solvers without Standard Inf-Sup Conditions

The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C^1 elements for velocity and C^0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> error estimate of the finite volume element methods on quadrilateral meshes

In this paper the optimal L ² error estimates of the finite volume element methods (FVEM) for Poisson equation are discussed on quadrilateral meshes. The trial function space is taken as isoparametric bilinear finite element space on quadrilateral partition, and the test function space is defined as piecewise constant space on dual partition. Under the assumption that all elements on quadrilateral meshes are O(h ²) quasi-parallel quadrilateral elements, we prove convergence rate to be O(h ²) in L ² norm.     

Efficient algorithms for solving tensor product finite element equations

Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC ⁰ quadratic orC ¹ cubic tensor product finite elements.     

Stability of penalty finite-element methods for nonconforming problems

Penalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth-order problems in which C¹-continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C¹-continuity constraint appears promising. In this study we demonstrate equivalence of the finite-element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.     

The Alexandroff Dimension of Digital Quotients of Euclidean Spaces

Alexandroff T ₀ -spaces have been studied as topological models of the supports of digital images and as discrete models of continuous spaces in theoretical physics. Recently, research has been focused on the dimension of such spaces. Here we study the small inductive dimension of the digital space X(W) constructed in [15] as a minimal open quotient of a fenestration W of R ⁿ . There are fenestrations of R ⁿ giving rise to digital spaces of Alexandroff dimension different from n , but we prove that if W is a fenestration, each of whose elements is a bounded convex subset of R ⁿ , then the Alexandroff dimension of the digital space X(W) is equal to n . Received December 6, 1999, and in revised form July 5, 2001, and August 31, 2001. Online publication January 7, 2002.     

Le genre de Mislin des espaces rationnellement équivalents à un produit de deux sphères de dimensions différentes

 Zabrodsky exact sequences are algebraic tools which express the genus set of a space X in term of its self-maps, when X has the rational homotopy type of a co-ℋ-space or an ℋ-space. Explicit examples show these methods can't be generalized to the class of all simply connected finite CW-complexes. We however construct a Zabrodsky exact sequence for those three cells CW-complexes rationally equivalent to the product of two spheres S ^k ×S ⁿ , n>k≥2. We deduce, from results of Morisugi-Oshima, the genus of some spherical bundles. Received: 17 March 2001 / Revised version: 8 August 2001     

An analysis for the compressible Stokes equations by first‐order system of least‐squares finite element method

This article applies the first‐order system least‐squares (fosls) finite element method developed by Cai, Manteuffel and McCormick to the compressible Stokes equations. By introducing a new dependent velocity flux variable, we recast the compressible Stokes equations as a first‐order system. Then it is shown that the ellipticity and continuity hold for the least‐squares functionals employing the mixture of H^?1 and L², so that the fosls finite element methods yield best approximations for the velocity flux and velocity. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:689–699, 2001     

On the stability of linear systems with fractional-order elements

Linear integer-order circuits are a narrow subset of rational-order circuits which are in turn a subset of fractional-order. Here, we study the stability of circuits having one fractional element, two fractional elements of the same order or two fractional elements of different order. A general procedure for studying the stability of a system with many fractional elements is also given. It is worth noting that a fractional element is one whose impedance in the complex frequency s-domain is proportional to s^α and α is a positive or negative fractional-order. Different transformations and methods will be illustrated via examples.     

On the complexity of a class of combinatorial optimization problems with uncertainty

We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements. We present a polynomial algorithm with the order of complexity O((min {p,m-p})² m) for the case where uncertainty is represented by means of interval estimates for the weights. We show that the problem is NP-hard in the case of an arbitrary finite set of possible scenarios, even if there are only two possible scenarios. This is the first known example of a robust combinatorial optimization problem that is NP-hard in the case of scenario-represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty. Received: July 1998 / Accepted: May 2000?Published online March 22, 2001     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001     

Distinct Distances in the Plane

It is shown that every set of n points in the plane has an element from which there are at least cn ^6/7 other elements at distinct distances, where c>0 is a constant. This improves earlier results of Erdős, Moser, Beck, Chung, Szemerédi, Trotter, and Székely. Received November 15, 2000, and in revised form December 13, 2000. Online publication April 6, 2001.     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

A comparison of methods of approximations for probabilities of death for fractions of a year

In this paper, a comparison of methods of approximating probabilities of death over fractions of a year is given. Previous and new methods in actuarial science and demography are considered. As the measure of distance the Kolmogorov statistic and L² are taken. The analysis was carried out using real data. Copyright © 2001 John Wiley & Sons, Ltd.     

On the super-approximation property of Galerkin's method with finite elements

Summary For Galerkin's method with finite elements as trial functions for strongly elliptic operator equations in the Hilbert scaleH ^t the super-approximation property and the optimal convergence rate are obtained by using the Aubin-Nitsche lemma. This applies in particular to spline collocation methods for a wide class of pseudodifferential equations.Dedicated to the memory of Professor Lothar Collatz     

Bouquet and join theorems for disentanglements

The disentanglement of certain augmentations is shown to be the topological join of a disentanglement and a Milnor fibre. The kth disentanglement of a finite map is defined and for corank 1 maps from ℂ ⁿ to ℂ ^{n +1} it is shown that they are homotopically equivalent to a wedge of spheres. Applications to the Mond conjecture are given. Oblatum 24-VII-2000 & 5-VII-2001?Published online: 12 October 2001     

Technical note: A fourth‐order finite difference scheme for a system of a 2D nonlinear elliptic partial differential equations

In this article we present a fourth‐order finite difference scheme, for a system of two‐dimensional, second‐order, nonlinear elliptic partial differential equations with mixed spatial derivative terms, using 13‐point stencils with a uniform mesh size h on a square region R subject to Dirichlet boundary conditions. The scheme of order h⁴ is derived using the local solution of the system on a single stencil. The resulting system of algebraic equations can be solved by iterative methods. The difference scheme can be easily modified to obtain formulae for grid points near the boundary. Computational results are given to demonstrate the performance of the scheme on some problems including Navier‐Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 43–53, 2001     

Adaptive stabilized mixed finite volume methods for the incompressible flow

In this article, we study adaptive stabilized mixed finite volume methods for the incompressible flows approximated using the lower order elements. A residual type of a posteriori error estimator is designed and studied with the derivation of upper and lower bounds between the exact solution and the finite volume solution. A discrete local lower bound between two successive finite volume solutions is also obtained. Also, convergence of the adaptive stabilized mixed finite volume methods is established. The presented methods have three prominent features. First, it is of practical convenience in real applications with the same partitions for velocity and pressure. Second, less computational time is required by easily applying both the lower order elements and the local grid refinement necessary for the elements of interest. Third, compared with the standard finite element method, its analysis of H¹‐norm and L²‐norm for the velocity and pressure are usually derived without any high order regularity conditions on the exact solution. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1424–1443, 2015