Models of two phase flows in porous media, used in petroleumengineering, lead to a system of two coupled equations with ellipticand parabolic degenerate terms, and two unknowns,the saturation and the pressure.For the purpose of their approximation, a coupled scheme, consisting ina finite volume method together witha phase-by-phase upstream weighting scheme, is used in the industrial setting.This paper presents a mathematical analysis of this coupled scheme, first showingthat it satisfies some a priori estimates:the saturation is shown to remain in a fixed interval, anda discrete L2(0,T;H1 (Ω)) estimate is proved for both the pressureand a function of the saturation. Thanks to these properties,a subsequence of the sequence of approximate solutions is shown toconverge to a weak solutionof the continuous equationsas the size of the discretization tends to zero.

We study in this paper some numerical schemes for hyperbolic systemswith unilateral constraint. In particular, we deal with the scalar case, the isentropicgas dynamics system and the full-gas dynamics system.We prove the convergence of the scheme to an entropy solutionof the isentropicgas dynamics with unilateral constraint on the density and mass loss.We also study the non-trivial steady states of the system.

This paper is concerned with the asymptotic behavior of thefinite difference solutions of a class of nonlinear reaction diffusion equations with time delay. By introducing a pair of coupled upper and lower solutions, an existence result of the solution is given and an attractor of the solution is obtained without monotonicity assumptions on the nonlinear reaction function. This attractor is a sector between two coupled quasi-solutions of the corresponding “steady-state" problem, which are obtained froma monotone iteration process. A sufficient condition, ensuring that two coupled quasi-solutions coincide, is given. Also given is the application to a nonlinear reaction diffusion problem with time delay for three different types of reaction functions, including some numerical results which validate the theoretical analysis.

We consider the time-harmonic eddy current problem in its electric formulationwhere the conductor is a polyhedral domain. By proving the convergencein energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11].We describe the singularities of the solutions.They are related to edge and corner singularities of certain problems for the scalarLaplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.

In this paper we propose a finite element method for the approximation ofsecond order elliptic problems on composite grids. The method isbased on continuous piecewise polynomial approximation on eachgrid and weak enforcement of the proper continuity at anartificial interface defined by edges (or faces) of one the grids.We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions,and present some numerical examples.

We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells in large displacements.

A current procedure that takes into account the Dirichlet boundary conditionwith non-smooth data is to change it into aRobin type condition by introducing a penalization term; a major effect of thisprocedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem wherethe control variable is the Dirichlet data.We describe the Robin penalization,and we bound the gap between the penalized and the non-penalized boundary controlsfor the small penalization parameter. Some numerical results are reported on to highlightthe reliability of such an approach.

Interest in meshfree methods in solving boundary-value problems has grownrapidly in recent years. A meshless method that has attracted considerableinterest in the community of computational mechanics is built around theidea of modified local Shepard's partition of unity. For these kinds ofapplications it is fundamental to analyze the order of the approximation inthe context of Sobolev spaces. In this paper, we study two differenttechniques for building modified local Shepard's formulas, and we provide atheoretical analysis for error estimates of the approximation in Sobolevnorms. We derive Jackson-type inequalities for h-p cloud functionsusing the first construction. These estimates are important in the analysisof Galerkin approximations based on local Shepard's formulas or h-pcloud functions.

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.

We propose and analyze a domain decomposition method on non-matching gridsfor partial differential equations with non-negativecharacteristic form. No weak or strong continuity of the finiteelement functions, their normal derivatives, or linearcombinations of the two is imposed across the boundaries of the subdomains.Instead, we employ suitable bilinear forms defined on the commoninterfaces, typical of discontinuous Galerkinapproximations.We prove an error bound which is optimal with respect to the mesh–size andsuboptimal with respect to the polynomial degree.Our analysis is valid for arbitrary shape–regular meshes and arbitrarypartitions into subdomains.Our method can be applied to advective, diffusive, and mixed–type equations,as well,and is well-suited for problems coupling hyperbolic and elliptic equations.We present some two-dimensionalnumerical results that support our analysis for the case of linear finite elements.

Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f(x)/x$ increases on [0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on $\xR_{+}^{2}$ with homogeneous Dirichlet boundary conditions by thesolution of $-\Delta u_{L}+f(u_{L})=0,$ on ]0,L[2 with adequatenon-homogeneous Dirichlet conditions. We show that the error uL - utends to zero exponentially fast, in the uniform norm.

We study in this paper the electromagnetic field generated in aconductor by an alternating current density. The resultinginterface problem (see Bossavit (1993)) between the metal and thedielectric medium is treated by a mixed–FEM and BEM couplingmethod. We prove that our BEM-FEM formulation is well posed andthat it leads to a convergent Galerkin method.

In this paper, we study the exterior boundary value problems of the Darwinmodel to the Maxwell's equations. The variational formulation is establishedand the existence and uniqueness is proved. We use the infinite element methodto solve the problem, only a small amount of computational work is needed.Numerical examples are given as well as a proof of convergence.

Bidomain models are commonly used for studying and simulatingelectrophysiological waves in the cardiac tissue. Most of thetime, the associated PDEs are solved using explicit finitedifference methods on structured grids. We propose an implicitfinite element method using unstructured grids for an anisotropicbidomain model. The impact and numerical requirements ofunstructured grid methods is investigated using a test casewith re-entrant waves.

We consider a non-conforming stabilized domaindecomposition technique forthe discretization of the three-dimensional Laplace equation.The aim is to extend the numerical analysis of residual error indicators tothis model problem. Two formulations of the problem are consideredand the error estimators are studied for both. In thefirst one, the error estimator provides upper and lower bounds forthe energy norm of the mortar finite element solution whereas inthe second case, it also estimates the error for the Lagrangemultiplier.

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow,and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship betweenthe regularized gradient flow (characterized by a small positive parameterε, see (1.7)) and the minimal surface flow [21]and the prescribed mean curvature flow [16].Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence tothe regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0in general cases.Provided that the regularized gradient flow problem possessesstrong solutions, which is proved possible if the datum functionsare regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the meshrelation k = O(h2). In particular, it is shown thatall error bounds depend on $\frac{1}{\varepsilon}$ onlyin some lower polynomial order for small ε.

Saccular aneurisms, swelling of a blood vessel,are investigated in order (i) to estimate the development risk ofthe wall lesion, before and after intravascular treatment,assuming that the pressure is the major factor,and (ii) to better plan medical interventions.Numerical simulations, using the finite element method,are performed in three-dimensional aneurisms.Computational meshes are derived from medical imaging datato take into account both between-subject and within-subjectanatomical variability of the diseased vessel segment.The 3D reconstruction is associated with a faceted surface.A geometrical model is then obtained to be finally meshed for a finite element use. The pulsatile flow of incompressible Newtonian blood is illustrated by numerical simulations carried out in two saccular aneurism types, a side- and a terminal-aneurism.High pressure zones are observed in the aneurism cavity,especially in the terminal one.

We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large.Two kinds of Zienkiewicz–Zhu (ZZ) type error estimators are derivedwhich originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient.The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained.The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh.The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.