The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws.The mathematical model is a set of six coupled differentialequations governing the time evolution of both the tumour cell population(cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunalenterocyte population, to be shielded from unwanted side effectsduring a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent freeplatinum Pt (the active drug) infusion law being given,the mathematical model allows to compute the time evolution of both tumour andvilli populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicityrhythm. The model has been derived using, as far as possible, experimental data.We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L ∞ criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.

We present a finite volume method based on the integration of the Laplaceequation on both the cells of a primal almost arbitrary two-dimensionalmesh and those of adual mesh obtained by joining the centers of the cells of the primal mesh.The key ingredient is the definition of discrete gradient and divergenceoperators verifying a discrete Green formula.This method generalizes an existing finite volume method thatrequires “Voronoi-type” meshes.We show the equivalence of this finite volume method with a non-conformingfinite element method with basis functions being P 1 on the cells,generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamond-cells, we prove a first-order convergence both in the $\xHone_0$ normand in the L² norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergencein the L² norm on general grids.They also indicate that this method performs particularly well for the approximationof the gradient of the solution, and may be used on degenerating triangular grids.An example of application on non-conforming locally refined grids is given.

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous,a static condensation procedure can becarried out, leading to a single-field nonconformingdiscretization scheme. For this latter formulation,we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis isdeveloped, proving first-order accuracy of the method in a discrete H 1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methodsfor 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 $\mathcal{O} (\Delta t^2 + h^{m + 1/2})$ where m is the degree of the local polynomials.

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

In this paper we solve the time-dependent incompressible Navier-Stokesequations by splitting the non-linearity and incompressibility, andusing discontinuous or continuous finite element methods in space. Weprove optimal error estimates for the velocity and suboptimalestimates for the pressure. We present some numerical experiments.

A Discontinuous Galerkin method is used for to thenumerical solution of the time-domain Maxwell equations onunstructured meshes. The method relies on the choice of local basisfunctions, a centered mean approximation for the surface integralsand a second-order leap-frog scheme for advancing in time. The methodis proved to be stable for cases with either metallic or absorbingboundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved formetallic cavities. Convergence is proved for $\mathbb{P}_k$ Discontinuous elements on tetrahedral meshes, as well as a discretedivergence preservation property. Promising numerical examples withlow-order elements show the potential of the method.

We consider the classical Interpolating Moving Least Squares (IMLS)interpolant as defined by Lancaster and Šalkauskas [Math. Comp.37 (1981) 141–158] andcompute the first and second derivative of this interpolant at the nodes of agiven grid with the help of a basic lemma on Shepard interpolants. We comparethe difference formulae with those defining optimal finite difference methods anddiscuss their deviation from optimality.

The aim of this paper is to present a method using both the ideas of sectionalapproach and moment methods in order to accurately simulate evaporationphenomena in gas-droplets flows. Using the underlying kinetic interpretation ofthe sectional method [Y. Tambour, Combust. Flame60 (1985)15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model.5 (2001) 537–572], we propose an extension of thisapproach based on a more accurate representation of the droplet size numberdensity in each section ensuring the exact conservation of two moments (asopposed to only one moment used in the classical approach). A correspondingsecond-order numerical scheme, with respect to space and droplet size variables,is also introduced and can be proved to be positive and to satisfy a maximumprinciple on the velocity and the mean droplet mass under a suitable CFL-likecondition. Numerical simulations have been performed and the results confirm theaccuracy of this new method even when a very coarse mesh for the droplet sizevariable (i.e.: a low number of sections) is used.

When analysing general systems of PDEs, it is important first to find the involutive form of the initial system.This is because the properties of the system cannot in general be determined if the system is not involutive.We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive formof the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution ofseveral flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.

This article discusses the numerical approximation oftime dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respectto a large Ginzburg-Landau parameter are established for asemi-discrete in time and a fully discrete approximationscheme. The proofs rely on an asymptotic expansion of the exact solution and a stability resultfor degree-one Ginzburg-Landau vortices. The error boundsprove that degree-one vortices can be approximated robustlywhile unstable higher degree vortices are critical.

The magnetization of a ferromagnetic sample solves anon-convex variational problem, where its relaxation by convexifyingthe energy density resolves relevantmacroscopic information. The numerical analysis of the relaxed modelhas to deal with a constrained convexbut degenerated, nonlocal energy functional in mixed formulation formagnetic potential u and magnetization m.In [C. Carstensen and A. Prohl, Numer. Math.90(2001) 65–99], the conforming P1 - (P0)d -element in d=2,3 spatialdimensions is shown to lead toan ill-posed discrete problem in relaxed micromagnetism, and suboptimalconvergence.This observation motivated anon-conforming finite element method which leads toa well-posed discrete problem, with solutions converging atoptimal rate.In this work, we provide both an a priori and a posteriori error analysis for twostabilized conforming methods which account for inter-element jumps of thepiecewise constant magnetization.Both methods converge at optimal rate;the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

This paper addresses some results on the development of an approximate methodfor computing the acoustic field scattered by a three-dimensional penetrable object immersed into an incompressiblefluid. The basic idea of the method consists in using on-surface differentialoperators that locally reproduce the interior propagation phenomenon. This approach leads tointegral equation formulations with a reduced computational cost compared to standard integral formulations couplingboth the transmitted and scattered waves. Theoreticalaspects of the problem and numerical experiments are reported to analyze the efficiency ofthe method and precise its validity domain.

We propose a general approach for the numerical approximation ofoptimal control problems governed by a linear advection–diffusionequation, based on a stabilization method applied to theLagrangian functional, rather than stabilizing the state andadjoint equations separately. This approach yields a coherentlystabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order termsare needless. Our a posteriori estimates stems from splitting theerror on the cost functional into the sum of an iteration errorplus a discretization error. Once the former is reduced below agiven threshold (and therefore the computed solution is “near”the optimal solution), the adaptive strategy is operated on thediscretization error. To prove the effectiveness of the proposedmethods, we report some numerical tests, referring to problems inwhich the control term is the source term of theadvection–diffusion equation.

We discuss the occurrence of oscillationswhen using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered andnon-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema.The dependence of oscillatory properties on the numerical viscositycoefficient is investigated rigorously for the LFt schemes, illuminating alsothe properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient areprone to oscillations at data extrema. For all LFt schemes except for the classicalLax-Friedrichs method, occurring oscillations are damped in the course of a computation.This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggeredversion may yield oscillatory results, while it can be shown rigorously that the staggered NTscheme does not produce oscillations when using the classical minmod-limiter under arestriction on the time step size. Note that this restriction is not thesame as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper.