Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

Existence and convergence for quasi‐static evolution in brittle fracture

This paper investigates the mathematical well‐posedness of the variational model of quasi‐static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford and Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time‐continuous quasi‐static growth is to pass to the limit as the time‐discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {u_n} of SBV‐functions to its BV‐limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u_n}. In particular, it is shown that the notion of minimizer of a Mumford and Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time‐continuous quasi‐static evolution. © 2003 Wiley Periodicals, Inc.     

Finite analytic numerical method for solving two‐dimensional quasi‐Laplace equation

A typical power series analytic solution of quasi‐Laplace equation in the infinitesimal angle domain around the singular point of the square cells is provided in this article. Toward the singular point, the gradient of the potential variable will tend to infinity, which is described by the first term of the power series solution. Based on this analytic solution, three finite analytic numerical methods are proposed. These methods are analogous and are constructed, respectively, when considering different numbers of the terms or using different schemes to determine the relevant parameters in the power series. Numerical examples show that all of the three finite analytic numerical methods proposed can provide rather accurate solutions than the traditional numerical methods. In contrast, when using the traditional numerical schemes to solve the quasi‐Laplace equation in a strong heterogeneous medium, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result. In practical applications, subdividing each origin cell into 2 × 2 or 3 × 3 subcells is enough for the finite analytical numerical methods to get relatively accurate results. The finite analytical numerical methods are also convenient to construct the flux field with high accuracy.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1755–1769, 2014     

Quasi‐Toeplitz splitting iteration methods for unsteady space‐fractional diffusion equations

We construct a class of quasi‐Toeplitz splitting iteration methods to solve the two‐sided unsteady space‐fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi‐circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving hyperbolic conservation laws using multiquadric quasi‐interpolation

In this article, we apply the univariate multiquadric (MQ) quasi‐interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi‐interpolation corresponding to periodic and inflow‐outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi‐interpolation to approximate the spatial derivative of the differential equation and a low‐order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one‐dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is ??(τ), where τ is the temporal step. We can improve the accuracy by using the high‐order quasi‐interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Disjoint quasi‐kernels in digraphs

A quasi‐kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvátal and Lovász proved that every digraph has a quasi‐kernel. Recently, Gutin et al. raised the question of which digraphs have a pair of disjoint quasi‐kernels. Clearly, a digraph has a pair of disjoint quasi‐kernels cannot contain sinks, that is, vertices of outdegree zero, as each such vertex is necessarily included in a quasi‐kernel. However, there exist digraphs which contain neither sinks nor a pair of disjoint quasi‐kernels. Thus, containing no sinks is not sufficient in general for a digraph to have a pair of disjoint quasi‐kernels. In contrast, we prove that, for several classes of digraphs, the condition of containing no sinks guarantees the existence of a pair of disjoint quasi‐kernels. The classes contain semicomplete multipartite, quasi‐transitive, and locally semicomplete digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:251‐260, 2008     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Radial basis collocation method and quasi‐Newton iteration for nonlinear elliptic problems

This work presents a radial basis collocation method combined with the quasi‐Newton iteration method for solving semilinear elliptic partial differential equations. The main result in this study is that there exists an exponential convergence rate in the radial basis collocation discretization and a superlinear convergence rate in the quasi‐Newton iteration of the nonlinear partial differential equations. In this work, the numerical error associated with the employed quadrature rule is considered. It is shown that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF. The combined errors due to radial basis approximation, quadrature rules, and quasi‐Newton and Newton iterations are also presented. This result can be extended to finite element or finite difference method combined with any iteration methods discussed in this work. The numerical example demonstrates a good agreement between numerical results and analytical predictions. The numerical results also show that although the convergence rate of order 1.62 of the quasi‐Newton iteration scheme is slightly slower than rate of order 2 in the Newton iteration scheme, the former is more stable and less sensitive to the initial guess. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Quasi‐graphic matroids*

Frame matroids and lifted‐graphic matroids are two interesting generalizations of graphic matroids. Here, we introduce a new generalization, quasi‐graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted‐graphic matroids, it is easy to certify that a 3‐connected matroid is quasi‐graphic. The main result is that every 3‐connected representable quasi‐graphic matroid is either a lifted‐graphic matroid or a frame matroid.     

Bases and quasi‐reflexivity in Fréchet spaces

A Fréchet space E is quasi‐reflexive if, either dim(E″/E) < ∞, or E″[β(E″,E′)]/E is isomorphic to ω. A Fréchet space E is totally quasi‐reflexive if every separated quotient is quasi‐reflexive. In this paper we show, using Schauder bases, that E is totally quasi‐reflexive if and only if it is isomorphic to a closed subspace of a countable product of quasi‐reflexive Banach spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi‐complements of the cappable degrees

Say that a nonzero c. e. degree b is a quasi‐complement of a c. e. degree a if a ∩ b = 0 and a ∪ b is high. It is well‐known (due to Shore) that each cappable degree has a high quasi‐complement. However, by the existence of the almost deep degrees, there are nonzero cappable degrees having no low quasi‐complements. In this paper, we prove that any nonzero cappable degree has a low₂ quasi‐complement. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coloring quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. A theorem of Shannon's implies that if G is a line graph, then it can be properly colored using no more than 3/2 ω(G) colors, where ω(G) is the size of the largest clique in G. In this article, we extend this result to all quasi‐line graphs. We also show that this bound is tight. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Single‐cell discretization of O(kh2 + h4) for ∂u/∂n for three‐space dimensional mildly quasi‐linear parabolic equation

In this article, using a single computational cell, we report some stable two‐level explicit finite difference approximations of O(kh² + h⁴) for ?u/?n for three‐space dimensional quasi‐linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x‐, y‐, and z‐coordinate axes, then ?u/?n at an internal grid point becomes ?u/?x, ?u/?y, and ?u/?z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 327–342, 2003.     

Infinite families of non‐embeddable quasi‐residual Menon designs

A Menon design of order h² is a symmetric (4h²,2h²‐h,h²‐h)‐design. Quasi‐residual and quasi‐derived designs of a Menon design have parameters 2‐(2h² + h,h²,h²‐h) and 2‐(2h²‐h,h²‐h,h²‐h‐1), respectively. In this article, regular Hadamard matrices are used to construct non‐embeddable quasi‐residual and quasi‐derived Menon designs. As applications, we construct the first two new infinite families of non‐embeddable quasi‐residual and quasi‐derived Menon designs. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 53–62, 2009     

An efficient method for fractional nonlinear differential equations by quasi‐Newton's method and simplified reproducing kernel method

An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi‐Newton's method, which is based on Fréchet derivative. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi‐Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.     

A singularly perturbed problem of photon transport in a time‐dependent region

The paper is devoted to the numerical study of a singularly perturbed transport linear integro‐differential equation, in a time‐dependent domain with slab geometry. After a brief summary on existence and uniqueness results for such a model, we test the error between the exact solution and its quasi‐static approximation, which satisfies a simpler equation. Copyright © 2002 John Wiley & Sons, Ltd.     

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.     

A note on triangle‐free quasi‐symmetric designs

Triangle‐free quasi‐symmetric 2‐ (v, k, λ) designs with intersection numbers x, y; 0<x<y<kand λ>1, are investigated. It is proved that λ?2y ? x ? 3. As a consequence it is seen that for fixed λ, there are finitely many triangle‐free quasi‐symmetric designs. It is also proved that: k?y(y ? x) + x. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:422‐426, 2011     

On the complexity of axiomatizations of the class of representable quasi‐polyadic equality algebras

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omegaUsing games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omega$ and $l< n,$ $k < n$, k′ < ω are natural numbers, then Σ contains infinitely equations in which ? occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim