Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.     

Convergence results for an accelerated nonlinear cimmino algorithm

Summary We present an accelerated version of Cimmino's algorithm for solving the convex feasibility problem in finite dimension. The algorithm is similar to that given by Censor and Elfving for linear inequalities. We show that the nonlinear version converges locally to a weighted least squares solution in the general case and globally to a feasible solution in the consistent case. Applications to the linear problem are suggested.     

The alpha interpolation method for the solution of an eigenvalue problem

We present a weighted residual finite element method for the solution of an eigenvalue problem. As a test function, we take a linear combination of two functions which belong to different spaces. We call this method the alpha interpolation method (AIM) for the eigenvalue problem. We compare the AIM with the Standard-Galerkin finite element method (SGFEM).     

On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions

We consider the Poisson equations for denumerable Markov chains with unbounded cost functions. Solutions to the Poisson equations exist in the Banach space of bounded real-valued functions with respect to a weighted supremum norm such that the Markov chain is geometrically ergodic. Under minor additional assumptions the solution is also unique. We give a novel probabilistic proof of this fact using relations between ergodicity and recurrence. The expressions involved in the Poisson equations have many solutions in general. However, the solution that has a finite norm with respect to the weighted supremum norm is the unique solution to the Poisson equations. We illustrate how to determine this solution by considering three queueing examples: a multi-server queue, two independent single server queues, and a priority queue with dependence between the queues.     

A modified method for constructing efficient solutions structure of MOLP

This paper deals with a recently proposed algorithm for obtaining all weak efficient and efficient solutions in a multi objective linear programming (MOLP) problem. The algorithm is based on solving some weighted sum problems, and presents an easy and clear solution structure. We first present an example to show that the algorithm may fail when at least one of these weighted sum problems has not a finite optimal solution. Then, the algorithm is modified to overcome this problem. The modified algorithm determines whether an efficient solution exists for a given MOLP and generates the solution set correctly (if exists) without any change in the complexity.     

D'Alembert formula on finite one-dimensional networks

We find a d'Alembert type formula for the solution of the Cauchy problem for the wave equation on finite weighted networks. We also discuss the periodicity in time of the solution in terms of the spectrum of the discrete graph associated with the network and finally we present two significant examples to illustrate and clarify the general analysis.     

SUPERCRITICAL SEMILINEAR WAVE EQUATION WITH NON-NEGATIVE POTENTIAL

We prove a weighted L ^∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Blow‐up phenomena for a semilinear parabolic equation with weighted inner absorption under nonlinear boundary flux

Blow‐up phenomena for a nonlinear divergence form parabolic equation with weighted inner absorption term are investigated under nonlinear boundary flux in a bounded star‐shaped region. We assume some conditions on weight function and nonlinearities to guarantee that the solution exists globally or blows up at finite time. Moreover, by virtue of the modified differential inequality, upper and lower bounds for the blow‐up time of the solution are derived in higher dimensional spaces. Three examples are presented to illustrate applications of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

A dual algorithm for the minimum covering weighted ball problem in {\mathbb{R}^n}

The nonlinear convex programming problem of finding the minimum covering weighted ball of a given finite set of points in ${\mathbb{R}^n}$ is solved by generating a finite sequence of subsets of the points and by finding the minimum covering weighted ball of each subset in the sequence until all points are covered. Each subset has at most n + 1 points and is affinely independent. The radii of the covering weighted balls are strictly increasing. The minimum covering weighted ball of each subset is found by using a directional search along either a ray or a circular arc, starting at the solution to the previous subset. The step size is computed explicitly at each iteration.     

Dirichlet Problems in Polyhedral Domains I: Regularity of the Solutions

The solution of the Dirichlet problem relative to an elliptic operator in a polyhedron has a complex singular behaviour near edges and vertices. Here we show that this solution and its conormal derivative have a global regularity in appropriate weighted Sobolev spaces. We also investigate some compact embeddings of these spaces. The present results will be applied in a forthcoming work to the constructive treatment of the problem by optimal convergent finite clement method and boundary element method.     

The finite element method for a boundary value problem with strong singularity

The existence and uniqueness of the R_ν-generalized solution for the third-boundary-value problem and the non-self-adjoint second-order elliptic equation with strong singularity are established. We construct a finite element method with a basis containing singular functions. The rate of convergence of the approximate solution to the R_ν-generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.     

非稳态奇异系数微分方程的时间间断时空有限元方法

针对一类二维单轴奇异系数非稳态问题构造了一种时间间断时空有限元格式,利用以Radau点为节点的Lagrange插值多项式的特性,结合有限差分法和有限元法的技巧证明了格式的稳定性和有限元解的时间最大模、空间加权L²（?）-模误差估计.最后列举了一些数值试验结果,验证了理论结果和格式的可行性.     

A new approach to the weighted peak-constrained least-square error FIR digital filter optimal design problem

This paper deals with the design of linear-phase finite impulse response (FIR) digital filters using weighted peak-constrained least-squares (PCLS) optimization. The PCLS error design problem is formulated as a quadratically constrained quadratic semi-infinite programming problem. An exchange algorithm with a new exchange rule is proposed to solve the problem. The algorithm provides the approximate optimal solution after a finite number of iterations. In particular, the subproblem solved at each iteration is a quadratically constrained quadratic programming. We can rewrite it as a conic optimization problem solvable in polynomial time. For illustration, numerical examples are solved using the proposed algorithm.     

Global existence and finite time blow up for the weighted semilinear wave equation

In this paper, we explain how weighted Strichartz estimates could be exploited to deal with the long time existence problem for the weighted semilinear wave equation with small data. When the solution blows up in finite time, we obtain the estimates for the lifespan of the solution.     

Solution of Maxwell equation in axisymmetric geometry by Fourier series decompostion and by use of H (rot) conforming finite element

Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry. Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally, following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element. Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999     

Galerkin and weighted Galerkin methods for a forward-backward heat equation

Summary. Galerkin and weighted Galerkin methods are proposed for the numerical solution of parabolic partial differential equations where the diffusion coefficient takes different signs. The approach is based on a simultaneous discretization of space and time variables by using continuous finite element methods. Under some simple assumptions, error estimates and some numerical results for both Galerkin and weighted Galerkin methods are presented. Comparisons with the previous methods show that new methods not only can be used to solve a wider class of equations but also require less regularity for the solution and need fewer computations. Received March 3, 1995     

带变系数的第二类奇异积分方程的Galerkin解法

基于对未知函数用适当的正交多项式进行逼近，本文讨论了带变系数的第二类奇异积分方程的Galerkin解法，证明了逼近解的存在唯一性，给出了逼近解在带权L^2模和一致模下的误差估计．     

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Chebyshev approximation of the null function by an affine combination of complex exponential functions

We describe the theoretical solution of an approximation problem that uses a finite weighted sum of complex exponential functions. The problem arises in an optimization model for the design of a telescope array occurring within optical interferometry for direct imaging in astronomy. The problem is to find the optimal weights and the optimal positions of a regularly spaced array of aligned telescopes, so that the resulting interference function approximates the zero function on a given interval. The solution is given by means of Chebyshev polynomials.