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.

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.

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 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 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 paper, a Dirichlet-Neumann substructuring domaindecomposition method is presented for a finite elementapproximation to the nonlinear Navier-Stokes equations. It isshown that the Dirichlet-Neumann domain decomposition sequenceconverges geometrically to the true solution provided the Reynoldsnumber is sufficiently small. In this method, subdomain problemsare linear. Other version where the subdomain problems are linearStokes problems is also presented.

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.