This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.

Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined...

The paper concerns an approximation of an eigenvalue problem for two forms on a Hilbert space $X$. We investigate some approximation methods generated by sequences of forms $a_n$ and $b_n$ defined on a dense subspace of $X$. The proof of convergence of the methods is based on the theory of the external approximation of eigenvalue problems. The general results are applied to Aronszajn’s method.

Let $x_{k+1}=T{x}_k+b$ be an iterative process for solving the operator equation $x=Tx+b$ in Hilbert space $X$. Let the sequence ${\left\{x_k\right\}}_{k=o}^{\infty }$ formed by the above described iterative process be convergent for some initial approximation $x_o$ with a limit $x^{*}=T{x}^{*}+b$. For given $l>1,m_0,m_1,\cdots ,m_l$ let us define a new sequence ${\left\{y_k\right\}}_{k=m_1}^{\infty }$ by the formula $y_k={\alpha }_0^{\left(k\right)}{x}_k+{\alpha }_1^{\left(k\right)}{x}_{k-m_1}+...+{\alpha }_l^{\left(k\right)}{x}_{k-m_l}$, where ${\alpha }_i^{\left(k\right)}$ are obtained by solving a minimization problem for a given functional. In this paper convergence properties of ${\alpha }_i^{\left(k\right)}$ are investigated and on the basis of the results thus obtainded it is proved that ${lim}_{k\to \infty }∥x^{*}-y_k∥/{∥x^{*}-x_k∥}^p=0$ for some $p\ge 1$.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$, we obtain error estimates in $L^2$ of order $𝒪(\Delta t^2+h^{m+1/2})$ where $m$ is the degree of the local polynomials.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$, we obtain error estimates in L2 of order $𝒪(\Delta t^2+h^{m+1/2})$ where m is the degree of the local polynomials.

Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges. A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations is emphasized....