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Let $ \mathcal P $ be the interior of a parabola of the complex plane. We show that, for all operators with numerical range contained in $ \mathcal P $ , $ \mathcal P $ is a 5-spectral set.

Let S be a strip or a sector of the complex plane. Being given a linear operator T on a Hilbert space whose numerical range is contained in S and a holomorphic function f uniformly bounded in S, we provide an estimate of the norm of f( T) with respect to this uniform bound.

This paper deals with the systems of algebraic equations arising in the application ofB-stable Runge-Kutta methods. It is shown that under natural assumptions such systems do not always have a solution. In addition, general sufficient conditions are presented under which such systems do have unique solutions.

Abstract: We analyze semidiscrete and fully discrete finite element approximations to the solution of an initial boundary value problem for a model ultraparabolic equation.

The stability of the $ {L_2}$, considered as a map in $ {L_p}$, $ 1 \leqslant p \leqslant \infty $.

Semidiscrete finite element methods for a semilinear parabolic equation in $ {R^d}$, were considered by Johnson, Larsson, Thomée, and Wahlbin. With h the discretization parameter, it was proved that, for compatible and bounded initial data in $ {H^\alpha }$ $ O({h^{2 + \alpha }})$t positive, and for $ \alpha = 0$ $ 0 \leq \alpha < 2$ , which is sharp.

We prove the analyticity (uniform in h) of the semigroups generated on $ {L_p}(0,1),1 \leq p \leq \infty $ of a one-dimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.

It is shown thatA-acceptable and, more generally,A(θ)-arational approximations of bounded analytic semigroups in Banach space are stable. The result applies, in particular, to the Crank-Nicolson method.

Summary We set aP1-type finite element method scheme to approximate one dimensional elliptic equations. We prove that an appropriate choice of numerical integration formula improves the classical error estimates by a super convergence result at the nodes of the mesh:O(h4) instead ofO(h2). This result is actually very cheap: indeed, the integration formula are easy (Simpson) and, moreover, the structure and the size of the linear system are the same as those of the classical scheme.

A mathematical analysis of the penalty method applied to the initial boundary value problem for the Stokes equations is presented. We prove a regularity result for the penalty equations by means of a spectral Galerkin method. Based upon this result the discretization in time (by a multistep integration formula) and in space (by a finite element method) is shown to give an optimal L2-error estimate.

The microscopic structure of thermotropic liquid crystals is experimentally investigated. The behavior of aromatic cores and aliphatic chains are analysed when defects are introduced in the mesophase as follows: to a liquid crystal having long aliphatic chains (40.8) is added one having shorter chains (40.2). An E.S.R. study is performed using two nitroxide probes: in smectic phases the first one is localized close to the aromatic cores while the other remains in the vicinity of aliphatic chains. In the nematic phases, when concentration in 40.2 varies, the orientational potential is found to be homogeneous because of the molecular translation diffusion. For high ordered smectic phases (SB) the potential does not depend on the concentration of 40.2 while introducing defects close to the aliphatic chains make the corresponding potential vary in a significant manner. In the SB phase die effect observed by E.S.R. (decrease of the order parameter of probe II) is more significant than the splitting variation previously observed by D.M.R. These results are consistent with a "partial rigidity" of aliphatic chains (interpenetrating structure of layers, intrinsic rigidity, etc.), limiting the new possible conformations in the mixture and with a localization of the probes corresponding to a low orientational potential (the "aliphatic density defects").