Abstract. We consider space semi-discretizations of the 1 − d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h → 0 this finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both finite-difference and finite-element semi-discretizations.Résumé. On considère l'approximation par différences finies etéléments finis en espace de l'équation des ondes 1 − d avec des conditions aux limites de Dirichlet homogènes. Onétudie le problème de l'observabilité frontière, i.e., la possibilité d'estimer l'énergie totale des solutions par l'énergie concentrée sur un extrême du bord, uniformement lorsque h, le pas de la discrétisation, tend vers zéro. On démontre que cette estimation uniforme n'a pas lieuà cause d'un comportement singulier des fonctions propresà hautes fréquences. Néanmoins, on démontre une estimation uniforme dans des sous-espaces convenables de solutions qui, lorsque h → 0, finissent par couvrir l'espace d'énergie tout entier. On retrouve donc la propriété d'observabilité, bien connue pour le système continu, comme la limite des estimations discrètes lorsque le pas en espace tend vers zéro.

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High Pressure (HP) Processing has turned out to be very effective in prolonging the shelf life of some food. This paper deals with the modelling and simulation of the effect of the combination of high pressure and thermal treatments on food processing, focusing on the inactivation of certain enzymes. The behavior and stability of the proposed models are checked by various numerical examples. Furthermore, various simplified versions of these models are presented and compared with each other in terms of accuracy and computational time. The models developed in this paper provide a useful tool to design suitable industrial equipments and optimize the processes.

We introduce a full approximation storage (FAS) multigrid algorithm to find the finite element solution for a class of nonlinear monotone elliptic problems. Since the solution of the problem is equivalent to minimize a strictly convex functional, we use a Polak-Ribiere conjugate gradient method as the nonlinear smoother in our algorithm. The advantage in so doing is that we do not have to calculate derivatives of operators. We prove local convergence of our algorithm and illustrate its performance by solving benchmark problems.

This paper deals with the problem of determining the time-dependent thermal diffusivity coefficient of a medium, when the evolution of the temperature in a part of it is known. Such situations arise in the context of food technology, when thermal processes at high pressures are used for extending the shelf life of the food, in order to preserve its nutritional and organoleptic properties (Infante et al 2009 On the Modelling and Simulation of High Pressure Processes and Inactivation of Enzymes in Food Engineering pp 2203-29 and Otero et al 2007 J. Food Eng. 78 1463-70). The phenomenon is modeled by the heat equation involving a term which depends on the source temperature and pressure increase, and appropriate initial and boundary conditions. We study the inverse problem of determining time-dependent thermal diffusivities k, when some temperature measurements at the border and inside the medium are known. We prove the uniqueness of the inverse problem solution under suitable a priori assumptions on regularity, size and growth of k.