In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients.The main unknowns of the system are the sediment thickness h, the L surface concentrations $c_i^s$ in lithology i of the sediments at the top of the basin, and the L concentrations c i in lithology i of the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for $c_i^s$ with a linear advection equation for c i for which $c_i^s$ appears as an input boundary condition. For this coupled system, a weak formulation is introduced which is shown to have a unique solution. An implicit finite volume scheme is derived for which we show stability estimates and the convergence to the weak solution of the problem.

We consider a strongly magnetized plasma described by a Vlasov-Poisson system with a large external magnetic field. The finite Larmor radius scaling allows to describe its behaviour at very fine scales. We give a new interpretation of the asymptotic equations obtained by Frénod and Sonnendrücker [SIAM J. Math. Anal. 32 (2001) 1227–1247] when the intensity of the magnetic field goes to infinity. We introduce the so-called polarization drift and show that its contribution is not negligible in the limit, contrary to what is usually said. This is due to the non linear coupling between the Vlasov and Poisson equations.

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in timestrong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence)of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$ . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as wellas the coupling of the local order of the constituent molecules of the membrane to its curvature.We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II7 (1997) 1509–1520; N. Uchida, Phys. Rev. E66 (2002) 040902] which replacesa Ginzburg-Landau penalization for the length of theorder parameter by a rigid constraint.We introduce a fully discrete scheme, consisting of piecewise linearfinite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence(up to subsequences) thereby proving the existence of a weak solutionto the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectricstructure mounted on a perfectly conducting plate in three dimensions. Given thatuniqueness of solution holds, existence of solution follows from a well-known Fredholmframework for the variational formulation of the problem in a suitable Sobolev space. Inthis paper, we derive a Rellich identity for a solution to this variational problem undersuitable smoothness conditions on the material parameter. Under additional non-trappingassumptions on the material parameter, this identity allows us to establish uniqueness ofsolution for all positive wave numbers.

We present equivalent conditions and asymptotic models for the diffraction problem ofelastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium.Due to the thinness of the layer with respect to the wavelength, this problem is wellsuited for the notion of equivalent conditions and the effect of the fluid medium on thesolid is as a first approximation local. We derive and validate equivalent conditions upto the fourth order for the elastic displacement. These conditions approximate theacoustic waves which propagate in the fluid region. This approach leads to solve onlyelastic equations. The construction of equivalent conditions is based on a multiscaleexpansion in power series of the thickness of the layer for the solution of thetransmission problem.

In this paper, a weighted regularization method for the time-harmonic Maxwell equationswith perfect conducting or impedance boundary condition in composite materials is presented.The computational domain Ω is the unionof polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularitiesnear geometric singularities, which are the interior and exterior edges and corners. The variational formulation of theweighted regularized problem is given on the subspace of ${\cal H}$ ( ${\bf curl}$ ;Ω)whose fields $\textit{\textbf{ u}}$ satisfy $w^\alpha$ div ( $\varepsilon{\textit{\textbf{u}}}$ )∈L2 (Ω) and have vanishing tangential traceor tangential trace in L2 ( $\partial\Omega$ ). The weight function $w(\bf x)$ is equivalentto the distance of $\bf x$ to the geometric singularities and the minimal weight parameter αis given in terms of the singular exponents of a scalar transmission problem.A density result is proven that guarantees the approximability of the solution field by piecewise regular fields.Numerical results for the discretization of the source problemby means of Lagrange Finite Elements of type P 1 and P 2 are given onuniform and appropriately refined two-dimensional meshes.The performance of the method in the case of eigenvalue problems is addressed.

This paper proposes a linear discrete-time scheme for general nonlinear cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger et al. [RAIRO Anal. Numer. 13 (1979) 297–312]. We analyze stability and convergence of the linear scheme. To this end, we apply the theory of reaction-diffusion system approximation. After discretizing the scheme in space, we obtain a versatile, easy to implement and efficient numerical scheme for the cross-diffusion systems. Numerical experiments are carried out to demonstrate the effectiveness of the proposed scheme.

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instancestreamline upwind Petrov-Galerkin (SUPG), are optimal on isotropicmeshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. Wedescribe the new methodand prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropicstabilization methods.The capability of the methodis illustrated by means of two numerical test problems.

We consider a family of shell finite elements with quadratic displacementsacross the thickness. These elements are very attractive, but compared to standard general shell elements they face another sourceof numerical locking in addition to shear and membrane locking. Thisadditional locking phenomenon – that we call “pinching locking” – is thesubject of this paper and we analyse a numerical strategy designed to overcomethis difficulty. Using a model problem in which only this specific sourceof locking is present, we are able to obtain error estimatesindependent of the thickness parameter, which shows that pinching lockingis effectively treated. This is also confirmed by some numerical experimentsof which we give an account.

The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperatureθ and an ODE with double obstaclesfor phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requiresthe stability constraint τ ≤ ε. Conforming piecewiselinear finite elements over highly graded simplicial mesheswith parameter h are further employed for space discretization.A posteriori errorestimates are derived for both unknowns θ and χ, whichexhibit the correct asymptotic order in terms of ε, h andτ. This result circumvents the use of duality, which does noteven apply in this context.Several numerical experiments illustrate the reliability of theestimators and document the excellent performance of the ensuingadaptive method.

Estimates for the combined effect of boundaryapproximation and numerical integration on the approximation of(simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficientsin convex domains with curved boundary by an isoparametric mixed finite element method, which,in the particular case of bending problems ofaniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending momenttensor field $\Psi= (\psi_{ij})_{1 \le i,j \le 2}$ anddisplacement field `u', have been developed.

In this paper, a general technique is developed to enlarge the velocity space ${\rm V}_h^1$ of the unstable -element by adding spaces ${\rm V}_h^2$ such thatfor the extended pair the Babuska-Brezzi condition is satisfied. Examplesof stable elements which can be derived in such a way imply the stability of the well-knownQ2/Q1 -element and the 4Q1/Q1 -element. However, our new elementsare much more cheaper. In particular, we shall see that more than half of theadditional degrees of freedom when switching from the Q 1 to the Q 2 and4Q1 , respectively, element are not necessary to stabilize theQ1/Q1 -element. Moreover, by using the technique of reduced discretizationsand eliminating the additional degrees of freedom we showthe relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilizedmethods for the Stokes and Navier-Stokes equations. Finally, we showhow the Brezzi-Pitkäranta stabilization and the SUPG method for theincompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

A new numerical method based on fictitious domain methods for shapeoptimization problems governed by the Poisson equation is proposed.The basic idea is to combine the boundary variation technique, in whichthe mesh is moving during the optimization, and efficient fictitiousdomain preconditioning in the solution of the (adjoint) state equations.Neumann boundary value problems are solved using an algebraic fictitiousdomain method. A mixed formulation based on boundary Lagrangemultipliers is used for Dirichlet boundary problems and the resultingsaddle-point problems are preconditioned with block diagonal fictitiousdomain preconditioners. Under given assumptions on the meshes, thesepreconditioners are shown to be optimal with respect to the conditionnumber. The numerical experiments demonstrate the efficiency ofthe proposed approaches.

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutionsof polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ $\mathbb{R}^d$ , d = 2 or 3, for the velocity andthe pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function $\beta^L(\cdot) :=\min(\cdot,L)$ in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker–Planck–Navier–Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions.We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms,and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classicalvariational approach without and with a regularization term in order tosmooth the contours of the classified image. Then we present the generaltopological asymptotic analysis, and we finally introduce its application tothe grey-level image classification problem.

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown.

The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependent damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings’ inherent difficulties.

This paper is concerned with the dual formulation of the interface problemconsisting of a linear partial differential equation with variable coefficientsin some bounded Lipschitz domain Ω in $\mathbb{R}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in theunbounded exterior domain Ωc := $\mathbb{R}^n\backslash\bar\Omega$ . The two problems are coupled by transmission andSignorini contact conditions on the interface Γ = ∂Ω. The exterior part of theinterface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequalitywith a linear operator. Then we treat the corresponding numerical scheme and discuss anapproximation of the NtD mapping with an appropriatediscretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximationproperties and a discrete inf-sup condition, we show unique solvability of the discrete scheme andobtain the corresponding a-priori error estimate. Next, we prove that these assumptions aresatisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdotf(x,u)=0$ on a closed Riemannian manifold M.For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$