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.

We examine the effect of numerical integration on the accuracy of high order conforming pyramidal finite element methods. Non-smooth shape functions are indispensable to the construction of pyramidal elements, and this means the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include non-smooth functions and show that, despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.

In this paper, we study some discretization schemes for diffusive flows in heterogeneous anisotropic porous media. We first introduce the notion of gradient scheme, and show that several existing schemes fall into this framework. Then, we construct two new gradient schemes which have the advantage of a small stencil. Numerical results obtained for real reservoir meshes show the efficiency of the new schemes, compared to existing ones.

The aim of this paper is to analyze a low order finite element methodfor a stiffened plate. The plate is modeled by Reissner-Mindlinequations and the stiffener by Timoshenko beams equations. Theresulting problem is shown to be well posed. In the case of concentricstiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysisand discretization of the first one is straightforward. The second oneis shown to have a solution bounded above and below independently of thethickness of the plate. A discretization based on DL3 finite elementscombined with ad-hoc elements for the stiffener is proposed.Optimal order error estimates are proved for displacements, rotationsand shear stresses for the plate and the stiffener. Numerical tests arereported in order to assess the performance of the method. Thesenumerical computations demonstrate that the error estimates areindependent of the thickness, providing a numerical evidence that themethod is locking-free.


A number of approaches for discretizing partial differential equations with random dataare based on generalized polynomial chaos expansions of random variables. These constitutegeneralizations of the polynomial chaos expansions introduced by Norbert Wiener toexpansions in polynomials orthogonal with respect to non-Gaussian probability measures. Wepresent conditions on such measures which imply mean-square convergence of generalizedpolynomial chaos expansions to the correct limit and complement these with illustrativeexamples.

In this article, we provide a priori error estimates for the spectral andpseudospectral Fourier (also called planewave) discretizations of theperiodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectraldiscretization of the periodic Kohn-Shammodel, within the local density approximation (LDA). These modelsallow to compute approximations of the electronic ground state energy and densityof molecular systems in the condensed phase. The TFW model is strictlyconvex with respect to the electronic density, and allows for acomprehensive analysis. This is not the case for the Kohn-Sham LDAmodel, for which the uniqueness of the ground state electronic densityis not guaranteed. We prove that, for any local minimizer $\Phi^0$ of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of $\Phi^0$ for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.

We analyze a nonlinear discrete scheme depending on second-order finite differences. Thisis the two-dimensional analog of a scheme which in one dimension approximates afree-discontinuity energy proposed by Blake and Zisserman as a higher-order correction ofthe Mumford and Shah functional. In two dimension we give a compactness result showingthat the continuous problem approximating this difference scheme is still defined onspecial functions with bounded hessian, and we give an upper and a lower bound in terms ofthe Blake and Zisserman energy. We prove a sharp bound by exhibiting thediscrete-to-continuous Γ-limit for a special class of functions, showingthe appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensionaleffect.

We study a two-phase pipe flow model with relaxation terms in the momentum and energy equations, driving the model towards dynamic and thermal equilibrium. These equilibrium states are characterized by the velocities and temperatures being equal in each phase. For each of these relaxation processes, we consider the limits of zero and infinite relaxation times. By expanding on previously established results, we derive a formulation of the mixture sound velocity for the thermally relaxed model. This allows us to directly prove a subcharacteristic condition; each level of equilibrium assumption imposed reduces the propagation velocity of pressure waves. Furthermore, we show that each relaxation procedure reduces the mixture sound velocity with a factor that is independent of whether the other relaxation procedure has already been performed. Numerical simulations indicate that thermal relaxation in the two-fluid model has negligible impact on mass transport dynamics. However, the velocity difference of sonic propagation in the thermally relaxed and unrelaxed two-fluid models may significantly affect practical simulations.

Numerically solving the Boltzmann kinetic equations with the small Knudsen number ischallenging due to the stiff nonlinear collision terms. A class of asymptotic-preservingschemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010)7625–7648] to handle this kind of problems. The idea is to penalize the stiff collisionterm by a BGK type operator. This method, however, encounters its own difficulty whenapplied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einsteinor Fermi-Dirac distribution) at each time step and every mesh point, one has to invert anonlinear equation that connects the macroscopic quantity fugacity with density andinternal energy. Setting a good initial guess for the iterative method is troublesome inmost cases because of the complexity of the quantum functions (Bose-Einstein orFermi-Dirac function). In this paper, we propose to penalize the quantum collision term bya ‘classical’ BGK operator instead of the quantum one. This is based on the observationthat the classical Maxwellian, with the temperature replaced by the internal energy, hasthe same first five moments as the quantum Maxwellian. The scheme so designed avoids theaforementioned difficulty, and one can show that the density distribution is still driventoward the quantum equilibrium. Numerical results are presented to illustrate theefficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also developa spectral method for the quantum collision operator.

We consider an incompressible flow problem in a N-dimensional fracturedporous domain (Darcy’s problem). The fracture is represented by a(N − 1)-dimensional interface, exchanging fluid with the surroundingmedia. In this paper we consider the lowest-order(ℝ T0, ℙ0) Raviart-Thomas mixed finite elementmethod for the approximation of the coupled Darcy’s flows in the porous media and withinthe fracture, with independent meshes for the respective domains. This is achieved thanksto an enrichment with discontinuous basis functions on triangles crossed by the fractureand a weak imposition of interface conditions. First, we study the stability andconvergence properties of the resulting numerical scheme in the uncoupled case, when theknown solution of the fracture problem provides an immersed boundary condition. We detailthe implementation issues and discuss the algebraic properties of the associated linearsystem. Next, we focus on the coupled problem and propose an iterative porousdomain/fracture domain iterative method to solve for fluid flow in both the porous mediaand the fracture and compare the results with those of a traditional monolithic approach.Numerical results are provided confirming convergence rates and algebraic propertiespredicted by the theory. In particular, we discuss preconditioning and equilibrationtechniques to make the condition number of the discrete problem independent of theposition of the immersed interface. Finally, two and three dimensional simulations ofDarcy’s flow in different configurations (highly and poorly permeable fracture) areanalyzed and discussed.

An optimal control problem governed by a bilinear elliptic equation is considered. Thisproblem is solved by the sequential quadratic programming (SQP) method in aninfinite-dimensional framework. In each level of this iterative method the solution oflinear-quadratic subproblem is computed by a Galerkin projection using proper orthogonaldecomposition (POD). Thus, an approximate (inexact) solution of the subproblem isdetermined. Based on a POD a-posteriori error estimator developed byTröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115]the difference of the suboptimal to the (unknown) optimal solution of the linear-quadraticsubproblem is estimated. Hence, the inexactness of the discrete solution is controlled insuch a way that locally superlinear or even quadratic rate of convergence of the SQP isensured. Numerical examples illustrate the efficiency for the proposed approach.