On Zienkiewicz–Zhu error estimators for Maxwell's equations

We consider a posteriori Zienkiewicz–Zhu (ZZ) type error estimators for the Maxwell equations. The main tool is the use of appropriate recovered values of the electric field and its curl. To cite this article: S. Nicaise, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the controllability for Maxwell's equations in specific media

In this Note we study the problem of exact controllability of the Maxwell's equations in specific media with two different models, on the one hand the so-called Drude–Born–Fedorov model, in the time domain, and on the other hand a simplified bilinear medium.For the first one we prove the non approximate controllability whereas for the second one we are able to prove the exact controllability under the usual conditions of the wave equation. To cite this article: P. Courilleau, T. Horsin, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Parallel summation,Maxwell's principle and the infimum of projections

In this paper we apply Maxwell's principle to give simple proofs of the properties of R: S, the parallel sum of two positive semi-definite linear operators. The parallel sum has been studied by Anderson, Ando, Duffin, Fillmore, Mitra, Williams, and others. In particular we give a short, elementary, and geometric proof of the result of Anderson and Duffin that gives the infimum of two orthogonal projections as twice their parallel sum.     

An error estimate for matrix equations

This paper proposes a new method for estimating the error in the solution of matrix equations. The estimate is based on the adjoint method in combination with small sample statistical theory. It can be implemented simply and is inexpensive to compute. Numerical examples are presented which illustrate the power and effectiveness of the new method.     

The weak null condition for Einstein's equations

We show that Einstein's equations of General Relativity expressed in wave coordinates satisfy a ‘weak null condition’. In a forthcoming article we will use this to prove a global existence result for Einstein's equations in wave coordinates with small initial data. To cite this article: H. Lindblad, I. Rodnianski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Constraint-Preserving Hybrid Finite Element Methods for Maxwell’s Equations

Foundations of Computational Mathematics - Maxwell’s equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux...     

An error estimate for Volterra integral equations

A bound is given for the error incurred by using Gregory's integration formula for solving nonlinear Volterra equations of the second kind.     

A hybrid numerical method for the shallow water equations with source term

In coastal oceanography there is interest in problems modeled by the shallow water equations, where variations in channel depth are accounted for by the presence of source terms. A numerical treatment for the solution of such problems is presented here, in terms of a hybrid approach, which combines a second-order TVD scheme for conservation law equations (assuming no source terms) with an eigenvector projection scheme that incorporates the effects of nonzero source terms (in regions where the bottom is not flat). For the case where an initially sharp wave profile is assumed, the progress of a wave as it traverses an estuary whose channel depth varies is calculated. Excellent numerical results are obtained. © 1996 John Wiley & Sons, Inc.     

Ważewski's Principle for retarded functional differential equations

Wa?ewski Principle is an important tool in the study of the asymptotic behavior of solutions of ordinary differential equations. A direct extension of this principle to retarded functional differential equations (RFDEs) can be obtained by noticing that solutions of RFDEs generate processes on C = C([?r, 0], Rⁿ) and by using the general version of Wa?ewski Principle for flows on topological spaces. The resulting method is of little use in applications, due to the infinite-dimensionality of the space C. This paper presents a “Razumikhin-type” extension of Wa?ewski's Principle, which is widely applicable to concrete examples. The main results are Corollaries 3.1 and 3.2. Also, an extension of the method to RFDEs with a merely continuous right-hand side is given, and a few examples illustrate the use of the method. Throughout the paper, a standard notation is used.     

Semilinear elliptic equations with Hardy potential and subcritical source term

Let \(\Omega \) be a smooth bounded domain in \({\mathbb {R}}^N\) (\(N>2\)) and \(\delta (x):=\text {dist}\,(x,\partial \Omega )\). Assume \(\mu \in {\mathbb {R}}_+, \nu \) is a nonnegative finite measure on \(\partial \Omega \) and \(g \in C(\Omega \times {\mathbb {R}}_+)\). We study positive solutions of

$$\begin{aligned} -\Delta u - \frac{\mu }{\delta ^2} u = g(x,u) \text { in } \Omega , \qquad \text {tr}^*(u)=\nu . \end{aligned}$$

(P)

Here \(\text {tr}^*(u)\) denotes the normalized boundary trace of u which was recently introduced by Marcus and Nguyen (Ann Inst H Poincaré Anal Non Linéaire, 34, 69–88, 2017). We focus on the case \(0<\mu < C_H(\Omega )\) (the Hardy constant for \(\Omega \)) and provide qualitative properties of positive solutions of (P). When \(g(x,u)=u^q\) with \(q>0\), we prove that there is a critical value \(q^*\) (depending only on \(N, \mu \)) for (P) in the sense that if \(q<q^*\) then (P) possesses a solution under a smallness assumption on \(\nu \), but if \(q \ge q^*\) this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical [see (1.28)]. We also investigate the case where g is linear or sublinear and give an existence result for (P).

     

An estimate for the remainder term of the asymptotics of the fundamental solution to parabolic equations with a small parameter

In this paper we deal with an asymptotics of the fundamental solution to parabolic equations of second order with a small parameter. We obtain a uniform multiplicative estimate of the difference between the fundamental solution and its asymptotics. We point out the asymptotics of the logarithmic limit in a special case when the matrix defining the principle term of the parabolic equation depends on the spatial variables and time.     

The first Darboux problem for wave equations with a nonlinear positive source term

We consider the first Darboux problem for nonlinear wave equations with positive power nonlinearity source term. Depending on the power of nonlinearity we investigate the problem on a global existence and blow-up of solutions of the first Darboux problem. The question of local solvability of the problem is also considered.     

Klee-Minty’s LP and upper bounds for Dantzig’s simplex method

Kitahara and Mizuno (2010) [2] get two upper bounds for the number of different basic feasible solutions generated by Dantzig’s simplex method. The size of the bounds highly depends on the ratio between the maximum and the minimum values of all the positive elements of basic feasible solutions. We show that the ratio for a simple variant of Klee-Minty’s LP is equal to the number of iterations by Dantzig’s simplex method for solving it.     

An extension of Gander’s result for quadratic equations

In the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with R-order of convergence at least four for quadratic equations.     

An asymptotic estimate related to Selberg's sieve

Suppose 1 ≤ z₁ ≤ z₂ ≤ N, and let $\text{Λ}{}_{\mathrm{i}}\text{(d) = μ(d)}\text{max(log}\text{(}\text{z}{}_{\mathrm{i}}\text{d}\text{), 0)}$ for i = 1, 2. We show that

$\text{∑}\text{N?n}\text{(}\text{∑}\text{d|n}\text{Λ}{}_1\text{(d))(}\text{∑}\text{e|n}\text{Λ}{}_1\text{(e)) = N}\text{log z}{}_1\text{+ O(N).}$

We then use this to improve a result of Barban-Vehov which has applications to zero-density theorems.     

An exact estimate result for a class of singular equations with critical exponents

We consider the singular boundary value problem