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This article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a space of initial conditions) to zero, in particular in the case when spectral condensation occurs. The purpose of the present article is to push forward the analysis to deal with any admissible control operator. The considered setting leads to applications to one dimensional parabolic-type equations or coupled systems of such equations. With such admissible control operator, the characterization of the minimal null control time is obtained thanks to the resolution of an auxiliary vectorial block moment problem (i.e. set in the control space) followed by a constrained optimization procedure of the cost of this resolution. This leads to essentially sharp estimates on the resolution of the block moment problems which are uniform with respect to the spectrum of the evolution operator in a certain class. This uniformity allows the study of uniform controllability for various parameter dependent problems. We also deduce estimates on the cost of controllability when the final time goes to the minimal null control time. We illustrate how the method works on a few examples of such abstract controlled systems and then we deal with actual coupled systems of one dimensional parabolic partial differential equations. Our strategy enables us to tackle controllability issues that seem out of reach by existing techniques. [ABSTRACT FROM AUTHOR]

The main concern of this article is to investigate the boundary controllability of some 2 × 2 one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components. [ABSTRACT FROM AUTHOR]

Recent progresses in intravital imaging have enabled highly-resolved measurements of periarteriolar oxygen gradients (POGs) within the brain parenchyma. POGs are increasingly used as proxies to estimate the local baseline oxygen consumption, which is a hallmark of cell activity. However, the oxygen profile around a given arteriole arises from an interplay between oxygen consumption and delivery, not only by this arteriole but also by distant capillaries. Integrating such interactions across scales while accounting for the complex architecture of the microvascular network remains a challenge from a modelling perspective. This limits our ability to interpret the experimental oxygen maps and constitutes a key bottleneck toward the inverse determination of metabolic rates of oxygen. We revisit the problem of parenchymal oxygen transport and metabolism and introduce a simple, conservative, accurate and scalable direct numerical method going beyond canonical Krogh-type models and their associated geometrical simplifications. We focus on a two-dimensional formulation, and introduce the concepts needed to combine an operator-splitting and a Green's function approach. Oxygen concentration is decomposed into a slowly-varying contribution, discretized by Finite Volumes over a coarse cartesian grid, and a rapidly-varying contribution, approximated analytically in grid-cells surrounding each vessel. Starting with simple test cases, we thoroughly analyze the resulting errors by comparison with highly-resolved simulations of the original transport problem, showing considerable improvement of the computational-cost/accuracy balance compared to previous work. We then demonstrate the model ability to flexibly generate synthetic data reproducing the spatial dynamics of oxygen in the brain parenchyma, with sub-grid resolution. Based on these synthetic data, we show that capillaries distant from the arteriole cannot be overlooked when interpreting POGs, thus reconciling recent measurements of POGs across cortical layers with the fundamental idea that variations of vascular density within the depth of the cortex may reveal underlying differences in neuronal organization and metabolic load. Author summary: The cerebral microvascular network is the logistics system that provides energy to brain cells at the right time and place. Blood flow and oxygen can now be observed dynamically in living rodents, which transformed our knowledge of the system and its role in ageing and disease. However, oxygen concentration at a given location is the result of a subtle balance between local cellular consumption, supply by neighboring vessels and their interconnections to distant ones. Thus, measurements are difficult to interpret without integrating this multi-scale component, which requires advanced computational models. This hinders our ability to bridge the gap between experiments in rodents and clinical applications in humans. In this work, we focus on oxygen transport between vessels, leveraging recent advances in multi-scale modelling and their mathematical foundations. By this way, we formulate for the first time a simple, conservative, accurate and scalable computational model for cerebral oxygen across scales, that is able to integrate the spatially heterogenous distribution of vessels. We illustrate how this model, combined to imaging, will pave the way towards better estimates of oxygen consumption, a hallmark of neural activity that cannot be directly measured. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems. [ABSTRACT FROM AUTHOR]

We study the upwind finite volume scheme on a general mesh for the initial- and boundary-value problem associated with a linear transport equation. For any bounded variation initial and boundary data we prove the (optimal) convergence rate 1/2 in the L¹-norm. Compared with previous works, our main contribution is to take into account the boundary data and to relax some regularity assumptions on the velocity field. As an intermediate result, we also provide a complete proof of the bounded variation regularity of weak solutions of such a general transport problem. [ABSTRACT FROM AUTHOR]

We establish a Poincaré-Wirtinger type inequality on some particular domains with a precise estimate of the constant depending only on the geometry of the domain. This type of inequality arises, for instance, in the analysis of finite volume (FV) numerical methods. As an application of our result, we prove uniform a priori bounds for the FV approximate solutions of the heat equation with Ventcell boundary conditions in the natural energy space defined as the set of those functions in $$H^1(\varOmega )$$ whose traces belong to $$H^1(\partial \varOmega )$$ . The main difficulty here comes from the fact that the approximation is performed on non-polygonal control volumes since the domain itself is non-polygonal. [ABSTRACT FROM AUTHOR]

"Discrete Duality Finite Volume" schemes (DDFV for short) on general 2D meshes, in particular, non-conforming ones, are studied for the Stokes problem with Dirichlet boundary conditions. The DDFV method belongs to the class of staggered schemes since the components of the velocity and the pressure are approximated on different meshes. In this paper, we investigate from a numerical and theoretical point of view, whether or not the stability condition holds in this framework for various kinds of mesh families. We obtain that different behaviors may occur depending on the geometry of the meshes. For instance, for conforming acute triangle meshes, we prove the unconditional Inf-Sup stability of the scheme, whereas for some conforming or nonconforming Cartesian meshes we prove that Inf-Sup stability holds up to a single unstable pressure mode. In any case, the DDFV method appears to be very robust. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new generalization of the well-known Cahn-Hilliard two-phase model for the modeling of n-phase mixtures. The model is derived using the consistency principle: we require that our n-phase model exactly coincides with the classical two-phase model when only two phases are present in the system. We give conditions for the model to be well-posed. We also present numerical results (including simulations obtained when coupling the Cahn-Hilliard system with the Navier-Stokes so as to obtain a phase-field model for multiphase flows) to illustrate the capability of such modeling. [ABSTRACT FROM AUTHOR]

In this paper we consider the boundary null controllability of a system of n parabolic equations on domains of the form Ω = (0, π) x Ω2 with Ω2 a smooth domain of ℝN--1, N >1. When the control is exerted on {0} x ϖ2 with ϖ2 ⊂Ω2, we obtain a necessary and sufficient condition that completely characterizes the null controllability. This result is obtained through the Lebeau--Robbiano strategy and requires an upper bound of the cost of the one-dimensional boundary null control on (0, π). The latter is obtained using the moment method and it is shown to be bounded by CeC/T when T goes to 0+. [ABSTRACT FROM AUTHOR]

This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initial-and boundary-value problems associated with the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in L∞ and the advection vector field v has low regularity properties, namely v ∈ L¹ (]0, T[, (W¹, ¹ (Q))d), with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in LP(Q), with p < +∞, of the approximate solution towards the unique weak solu-tion of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory. Note that this result remains valid if the data are suitably approximated in L¹. This is nothing but the discrete coun-terpart of the nice compactness properties deduced from the renormalized solution theory. We conclude with some numerical experiments showing that the convergence rate seems to be 1/2, as in the case of smoother advection fields, but this is still an open question up on up [ABSTRACT FROM AUTHOR]

In this article, we investigate numerical schemes for solving a three component Cahn-Hilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the free energy and thus the stability of the method. We study three different schemes and prove existence and convergence theorems. Theoretical results are illustrated by various numerical examples showing that the new semi-implicit discretization that we propose seems to be a good compromise between robustness and accuracy. [ABSTRACT FROM AUTHOR]

We propose a nonoverlapping Schwarz algorithm for solving ‘discrete duality finite-volume’ (DDFV) schemes on general meshes. In order to handle this problem the first step is to propose and study a convenient DDFV scheme for anisotropic elliptic problems with mixed Dirichlet/Fourier boundary conditions. Then we are able to build the corresponding Schwarz algorithm and to prove its convergence to the solution of the DDFV scheme on the initial domain. We finally give some numerical results in the case where the Schwarz iterations are used both as a solver and as a preconditioner. [ABSTRACT FROM PUBLISHER]

Abstract: We derive a semi-discrete two-dimensional elliptic global Carleman estimate, in which the usual large parameter is connected to the one-dimensional discretization step-size. The discretizations we address are some families of smoothly varying meshes. As a consequence of the Carleman estimate, we derive a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano, in the case of a discrete elliptic operator in one dimension. Here, this inequality concerns the lower part of the discrete spectrum. The range of eigenvalues/eigenfunctions we treat is however quasi-optimal and represents a constant portion of the discrete spectrum. For the associated parabolic problem, we then obtain a uniform null controllability result for this lower part of the spectrum. Moreover, with the control function that we construct, the -norm of the final state converges to zero super-algebraically as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced. [Copyright &y& Elsevier]

In this paper we study the approximation of solutions to linear and nonlinear elliptic-problems with discontinuous coefficients in the discrete duality finite volume (DDFV) framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems. [ABSTRACT FROM AUTHOR]

We consider finite-volume schemes on rectangular meshes for the p-Laplacian with Dirichlet boundary conditions. In Andreianov et al. (2004a, Math. Model. Numer. Anal., 38, 931–959), we constructed a family of schemes and proved discrete W1, p error estimates in the case of W2, p solutions of the homogeneous problem. Here we improve these estimates in the case of W4, 1 solutions on uniform meshes for p > 3, using symmetry properties of the schemes. The proof also works for the Laplace equation, giving O(R2) convergence for a family of nine-point finite-volume schemes. With the same ideas, using the improved coercivity inequalities of Barrett and Liu, we obtain even better W1, p, W1, 1 and L∞ convergence rates for special classes of regular solutions to the inhomogeneous problem—in particular, for solutions without critical points in Ω¯, for all p ∈ (1, ∞). Numerical examples are given. They suggest the optimality of the L∞ estimates, of order h2, obtained for solutions without critical points. [ABSTRACT FROM PUBLISHER]

In this paper we are concerned with the numerical study of viscoelastic diphasic flows.We first propose a complete model and give some formal justifications including the case where both or one phase is a non-Newtonian one. Then, we have implemented a numerical method taking into account the Lagrangian transport in the resolution of the extra-stress equation of Oldroyd type. We describe this fully-practical method and prove its stability so that one can perform long-time numerical simulations.Several significant tests (channel under shear, filling of tanks, breakup of jets...) illustrate the pertinency of the model and of the numerical method. We show that the results are in accordance with the various qualitative behaviors observed in physical experiments. [Copyright &y& Elsevier]

We consider a diphasic system in a high shear regime when separation of phases occurs. One can observe that the two phases organize themselves into numerous bands, parallel to the flow direction. Mathematically, these are solutions of a certain system of equations depending only upon the transversal variable (1D solutions). We study the stability of these 1D solutions with respect to 2D perturbations. The mathematical model used in our analysis is a coupling between a Cahn–Hilliard equation and the Navier–Stokes equations in two dimensions. We show that a small 2D perturbation of a given 1D solution persists for significant times. We give the precise size of such a perturbation and its time of persistence. Moreover, we obtain an asymptotic expansion of the solution in the considered cases. Note that, for a mathematical model to be realistic one has to take into account the fact that in experiments the high shear regime is obtained in elongated domains (a very thin Couette cell for example). Therefore, we perform the mathematical analysis of this problem in a stretched domain. [ABSTRACT FROM AUTHOR]

In this paper we study the coupling of the Navier–Stokes equations and the Cahn–Hilliard equation which stands for a model of a multi‐phase fluid under shear. We first study existence and uniqueness of solutions of the system in dimension 2 and 3 even if the diffusion coefficient is allowed to degenerate. In the last part, an asymptotic stability result is shown. [ABSTRACT FROM AUTHOR]