The present paper deals with a finite element approximation of partial differential equations when the domain is decomposed into sub-domains which are meshed independently. The method we obtain is never conforming because the continuity constraints on the boundary of the sub-domains are not imposed strongly but only penalized. We derive a selection rule for the penalty parameter which ensures a quasi-optimal convergence.

We consider optimal control problems for the bidomain equations of cardiacelectrophysiology together with two-variable ionic models, e.g. theRogers–McCulloch model. After ensuring the existence of global minimizers, we provide arigorous proof for the system of first-order necessary optimality conditions. The proof isbased on a stability estimate for the primal equations and an existence theorem for weaksolutions of the adjoint system.

Cultivating oleaginous microalgae in specific culturing devices such as raceways is seenas a future way to produce biofuel. The complexity of this process coupling non linearbiological activity to hydrodynamics makes the optimization problem very delicate. Thelarge amount of parameters to be taken into account paves the way for a usefulmathematical modeling. Due to the heterogeneity of raceways along the depth dimensionregarding temperature, light intensity or nutrients availability, we adopt a multilayerapproach for hydrodynamics and biology. For free surface hydrodynamics, we use amultilayer Saint–Venant model that allows mass exchanges, forced by a simplifiedrepresentation of the paddlewheel. Then, starting from an improved Droop model thatincludes light effect on algae growth, we derive a similar multilayer system for thebiological part. A kinetic interpretation of the whole system results in an efficientnumerical scheme. We show through numerical simulations in two dimensions that ourapproach is capable of discriminating between situations of mixed water or calm andheterogeneous pond. Moreover, we exhibit that a posteriori treatment ofour velocity fields can provide lagrangian trajectories which are of great interest toassess the actual light pattern perceived by the algal cells and therefore understand itsimpact on the photosynthesis process.

We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear)interface problem for the 2D Laplacian. We introduce some new a posteriorierror estimators based on the (h − h/2)-errorestimation strategy. In particular, these include the approximation error for the boundarydata, which allows to work with discrete boundary integral operators only. Using theconcept of estimator reduction, we prove that the proposed adaptive algorithm isconvergent in the sense that it drives the underlying error estimator to zero. Numericalexperiments underline the reliability and efficiency of the considered adaptivemesh-refinement.

This work aims at introducing modelling, theoretical and numerical studies related to a new downscaling technique applied to computational fluid dynamics.Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor.The local model, compatible with the Navier-Stokes equations, is usedfor the small scale computation (downscaling) of the consideredfluid. It isinspired by Pope's works on turbulence, and consists in a so-called Langevin system of stochastic differential equations. We introduce this model and exhibit its links with classical RANS models. Well-posedness, as well as mean-field interacting particle approximations and boundary condition issues are addressed. We present the numerical discretization of the stochastic downscaling method and investigate the accuracy of the proposed algorithm on simplified situations.

We prove the convergence of a finite volumemethod for a noncoercive linear elliptic problem, with right-handside in the dual space of the natural energy space of the problem.

In this work we derive a posteriori error estimates basedon equations residuals for the heat equation with discontinuousdiffusivity coefficients. The estimates are based on a fully discretescheme based on conforming finite elements in each time slab andon the A-stable θ-scheme with 1/2 ≤ θ ≤ 1.Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easyto identify a time-discretization error-estimatorand a space-discretization error-estimator. In this work we introduce a similarsplitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math.72 (1996) 313–348; Petzoldt, Adv. Comput. Math.16 (2002) 47–75] we have upper and lower boundswhose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the coupling with the ion acoustic waves which has to be taken into account to model the so called Brillouin instability. Here, besides the macroscopic density and the velocity of the plasma, one has to handle the space-time envelope of the main laser wave, the space-time envelope of the stimulated Brillouin backscattered laser wave and the space envelope of the Brillouin ion acoustic waves. Numerical methods are also described to deal with the paraxial model and the three-wave coupling system related to the Brillouin instability.

In several practically interesting applications of electromagnetic scattering theory like, e.g., scattering from small point-like objects such as buried artifacts or small inclusions in non-destructive testing, scattering from thin curve-like objects such as wires or tubes, or scattering from thin sheet-like objects such as cracks, the volume of the scatterers is small relative to the volume of the surrounding medium and with respect to the wave length of the applied electromagnetic fields.This smallness typically causes problems when solving direct scattering problems due to the need to discretize the objects and also when solving inverse scattering problems because small objects have very little effect on electromagnetic fields. In this paper we consider an asymptotic representation formula for scattered electromagnetic waves caused by low volume objects contained in some otherwise homogeneous three-dimensional bounded domain, assuming only that the scatterers are measurable and well-separated from the boundary of the domain.The formula yields a very general asymptotic model for electromagnetic scattering due to low volume objects that can either be used to simulate the corresponding electromagnetic fields or as the foundation of efficient reconstruction methods for inverse scattering problems with low volume scatterers.Our analysis extends results originally obtained in [Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal. 37 (2003) 159–173] for steady state voltage potentials to time-harmonic Maxwell's equations.

We present an integral equation method for solving boundary valueproblems of the Helmholtz equation in unbounded domains. Themethod relies on the factorisation of one of the Calderón projectors by an operator approximating the exterioradmittance (Dirichlet to Neumann) operator of the scatteringobstacle. We show how the pseudo-differential calculus allows usto construct such approximations and that this yields integralequations without internal resonances and being well-conditionedat all frequencies. An implementation technique is elaborated,where again reasonings from pseudo-differential calculus play animportant rôle. Some numerical examples are presented which appearto confirm that the new integral equation leads to linear systemswhich are much better conditioned than the classical ("direct")integral equations and hence have much better behaviour whensolved with iterative techniques and matrix sparsification.

In this article we study discontinuous Galerkin finite element discretizations of linearsecond-order elliptic partial differential equations with Dirac delta right-hand side. Inparticular, assuming that the underlying computational mesh is quasi-uniform, we derive ana priori bound on the error measured in terms of theL2-norm. Additionally, we develop residual-based aposteriori error estimators that can be used within an adaptive mesh refinementframework. Numerical examples for the symmetric interior penalty scheme are presentedwhich confirm the theoretical results.

In this paper we consider a model shape optimization problem. The state variable solvesan elliptic equation on a domain with one part of the boundary described as the graph of acontrol function. We prove higher regularity of the control and develop a priorierror analysis for the finite element discretization of the shape optimizationproblem under consideration. The derived a priori error estimates areillustrated on two numerical examples.

Intracellular transport in eukarya is attributed to motor proteins that transduce chemical energy into directed mechanical energy. This suggests that, in nonequilibrium systems, fluctuations may be oriented or organized to do work. Here we seek to understand how this is manifested by quantitative mathematical portrayals of these systems.

We introduce a phenomenological model for anti-angiogenic therapy in the treatment of metastatic cancers. It is a structured transport equation with a nonlocal boundary condition describing the evolution of the density of metastases that we analyze first at the continuous level. We present the numerical analysis of a lagrangian scheme based on the characteristics whose convergence establishes existence of solutions. Then we prove an error estimate and use the model to perform interesting simulations in view of clinical applications.

This paper discusses analytical and numerical issues related toelliptic equations with random coefficients which are generallynonlinear functions of white noise. Singularity issues are avoidedby using the Itô-Skorohod calculus to interpret the interactionsbetween the coefficients and the solution. The solution is constructedby means of the Wiener Chaos (Cameron-Martin) expansions. Theexistence and uniqueness of the solutions are established underrather weak assumptions, the main of which requires only that theexpectation of the highest order (differential) operator is anon-degenerate elliptic operator. The deterministic coefficientsof the Wiener Chaos expansion of the solution solve a lower-triangularsystem of linear elliptic equations (the propagator). This structureof the propagator insures linear complexity of the related numericalalgorithms. Using the lower triangular structure and linearity of thepropagator, the rate of convergence is derived for a spectral/hp finiteelement approximation. The results of related numerical experiments arepresented.

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of ${\mathbb R}^2$ with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary.Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous Neumann conditions, the emphasis is placed on transparent boundary conditions, which allow the computationof the solutions in the subdomains obtained by stopping the geometric construction after a finite number of steps. The proposed methods and algorithmswill be used numerically in forecoming papers.

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm thesuperconvergence property and suggest that it also holds for the lowest orderBrezzi-Douglas-Marini approximation.

We prove convergence and quasi-optimal complexity of an adaptive finite element algorithm on triangular meshes with standard mesh refinement. Our algorithm is based on an adaptive marking strategy. In each iteration, a simple edge estimator is compared to an oscillation term and the marking of cells for refinement is done according to the dominant contribution only. In addition, we introduce an adaptive stopping criterion for iterative solution which compares an estimator for the iteration error with the estimator for the discretization error.

The paper is devoted to the computation of two-phase flows in a porous mediumwhen applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them correspondingto solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme,are shown to give wrong approximations in some cases involving sharp porous profiles.The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal. 43 (2006) 796–824]for the computation of single phase flows in varying cross section ducts,provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L 1 norm of the error.

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.