In this paper, we are interested in modelling the flow of the coolant (water) in anuclear reactor core. To this end, we use a monodimensional low Mach number modelsupplemented with the stiffened gas law. We take into account potential phase transitionsby a single equation of state which describes both pure and mixture phases. In someparticular cases, we give analytical steady and/or unsteady solutions which providequalitative information about the flow. In the second part of the paper, we introduce twovariants of a numerical scheme based on the method of characteristics to simulate thismodel. We study and verify numerically the properties of these schemes. We finally presentnumerical simulations of a loss of flow accident (LOFA) induced by a coolant pump tripevent.

We consider an initial-boundary value problem for a generalized 2D time-dependentSchrödinger equation (with variable coefficients) on a semi-infinite strip. For theCrank–Nicolson-type finite-difference scheme with approximate or discrete transparentboundary conditions (TBCs), the Strang-type splitting with respect to the potential isapplied. For the resulting method, the unconditional uniform in time L2-stability isproved. Due to the splitting, an effective direct algorithm using FFT is developed now toimplement the method with the discrete TBC for general potential. Numerical results on thetunnel effect for rectangular barriers are included together with the detailed practicalerror analysis confirming nice properties of the method.

We examine the dynamics of fermentation process in a yeast cell. Our investigation focuses on the main branch pathway: pyruvate and acetaldehyde branch points. We formulate the kinetics for all enzymatic reactions as Michaelis–Menten models. Since the activity of an enzyme mainly depends on the conformational changes of the enzyme structure, the enzyme requires a certain period of time to reset its structure, until it is ready to bind substrates again. For this situation, a rate-limiting step exists, for which the catalytic process suffers a delay. Since all conversion processes are catalysed by enzymes, each reaction can experience a delay at a different time. To investigate how the delay affects the reaction processes, especially at the branch points, we propose that the rate-limiting step takes place at the first reaction. For this reason, a discrete time delay is introduced to the first kinetic model. We find a bifurcation diagram for the delay that depends on the rate of glucose supply and kinetic parameters of the first enzyme. By comparison, our analysis agrees with the numerical solution. Our numerical simulations also show that there is a certain glucose supply that may optimize ethanol production.

Although variance swaps have become an important financial derivative to hedge against volatility risks, closed-form formulae have been developed only recently, when the realized variance is defined on discrete sample points and no continuous approximation is adopted to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. In this paper, a new closed-form pricing formula for the value of a discretely sampled variance swap is presented under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model. With the newly found analytical formula, not only can all the hedging ratios of a variance swap be analytically derived, the numerical values of the swap price can be efficiently computed as well.

Stokes’ axisymmetrical translational motion of a slip sphere, located anywhere on the diameter of a virtual spherical fluid ‘cell’, is investigated. The fluid is micropolar and flows are parallel to the line connecting the two centres. An infinite-series solution is presented for the stream function, pressure field, vorticity, microrotation component, shear stress and couple stress of the flow. Basset-type slip boundary conditions on the sphere surface are used for velocity and microrotation. The Happel and Kuwabara boundary conditions are used on the fictitious surface of the cell model. Numerical results for the normalized drag force acting on the sphere are obtained with excellent convergence for various values of the volume fraction, the relative distance between the centre of the sphere and the virtual envelope, the vortex viscosity parameter and the slip coefficients of the sphere. In the special case when the spherical particle is in the concentric position with the cell surface, the numerical values of the normalized drag force agree with the available values in the literature.

We consider the efficient and reliable solution of linear-quadratic optimal controlproblems governed by parametrized parabolic partial differential equations. To this end,we employ the reduced basis method as a low-dimensional surrogate model to solve theoptimal control problem and develop a posteriori error estimationprocedures that provide rigorous bounds for the error in the optimal control and theassociated cost functional. We show that our approach can be applied to problems involvingcontrol constraints and that, even in the presence of control constraints, the reducedorder optimal control problem and the proposed bounds can be efficiently evaluated in anoffline-online computational procedure. We also propose two greedy sampling procedures toconstruct the reduced basis space. Numerical results are presented to confirm the validityof our approach.

In this article we develop a posteriori error estimates for second orderlinear elliptic problems with point sources in two- and three-dimensional domains. Weprove a global upper bound and a local lower bound for the error measured in a weightedSobolev space. The weight considered is a (positive) power of the distance to the supportof the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theoryhinges on local approximation properties of either Clément or Scott–Zhang interpolationoperators, without need of modifications, and makes use of weighted estimates forfractional integrals and maximal functions. Numerical experiments with an adaptivealgorithm yield optimal meshes and very good effectivity indices.

This work studies the heat equation in a two-phase material with spherical inclusions.Under some appropriate scaling on the size, volume fraction and heat capacity of theinclusions, we derive a coupled system of partial differential equations governing theevolution of the temperature of each phase at a macroscopic level of description. Thecoupling terms describing the exchange of heat between the phases are obtained by usinghomogenization techniques originating from [D. Cioranescu, F. Murat, Collège de FranceSeminar, vol. II. Paris 1979–1980; vol. 60 of Res. Notes Math. Pitman,Boston, London (1982) 98–138].

We prove Hölder regularity of the gradient, up to the boundary for solutions of somefully-nonlinear, degenerate elliptic equations, with degeneracy coming from thegradient.

In this paper, we present and study a mixed variational method in order to approximate,with the finite element method, a Stokes problem with Tresca friction boundary conditions.These non-linear boundary conditions arise in the modeling of mold filling process bypolymer melt, which can slip on a solid wall. The mixed formulation is based on adualization of the non-differentiable term which define the slip conditions. Existence anduniqueness of both continuous and discrete solutions of these problems is guaranteed bymeans of continuous and discrete inf-sup conditions that are proved. Velocity and pressureare approximated by P1 bubble-P1 finite element and piecewise linearelements are used to discretize the Lagrange multiplier associated to the shear stress onthe friction boundary. Optimal a priori error estimates are derived usingclassical tools of finite element analysis and two uncoupled discrete inf-sup conditionsfor the pressure and the Lagrange multiplier associated to the fluid shear stress.

For a stationary Markov process the detailed balance condition is equivalent to thetime-reversibility of the process. For stochastic differential equations (SDE’s), the timediscretization of numerical schemes usually destroys the time-reversibility property.Despite an extensive literature on the numerical analysis for SDE’s, their stabilityproperties, strong and/or weak error estimates, large deviations and infinite-timeestimates, no quantitative results are known on the lack of reversibility of discrete-timeapproximation processes. In this paper we provide such quantitative estimates by using theconcept of entropy production rate, inspired by ideas from non-equilibrium statisticalmechanics. The entropy production rate for a stochastic process is defined as the relativeentropy (per unit time) of the path measure of the process with respect to the pathmeasure of the time-reversed process. By construction the entropy production rate isnonnegative and it vanishes if and only if the process is reversible. Crucially, from anumerical point of view, the entropy production rate is an a posterioriquantity, hence it can be computed in the course of a simulation as the ergodicaverage of a certain functional of the process (the so-called Gallavotti−Cohen (GC) action functional). We computethe entropy production for various numerical schemes such as explicit Euler−Maruyama and explicit Milstein’s forreversible SDEs with additive or multiplicative noise. In addition we analyze the entropyproduction for the BBK integrator for the Langevin equation. The order (in thetime-discretization step Δt) of the entropy production rate provides a tool toclassify numerical schemes in terms of their (discretization-induced) irreversibility. Ourresults show that the type of the noise critically affects the behavior of the entropyproduction rate. As a striking example of our results we show that the Euler scheme formultiplicative noise is not an adequate scheme from a reversibilitypoint of view since its entropy production rate does not decrease withΔt.

We consider a model for flow in a porous medium with a fracture in which the flow in thefracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed byDarcy’s law. We give an appropriate mixed, variational formulation and show existence anduniqueness of the solution. To show existence we give an analogous formulation for themodel in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We showexistence and uniqueness of the solution and show that the solution for the model withDarcy’s law in the matrix is the weak limit of solutions of the model with theDarcy−Forchheimerlaw in theentire domain when the Forchheimer coefficient in the matrix tends toward zero.

We study properties of the functional

\begin{eqnarray} \mathscr{F}_{{\rm loc}}(u,\Omega):=\inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(\nabla u_{j})\udx\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rmloc}}^{1,r}\left(\Omega, \RN\right) \\ & u_{j}\tostar u\,\,\textrm{in}\BV\left(\Omega, \RN\right) \end{array} \right. \bigg\}, \end{eqnarray}$\begin{array}{ccc}{F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{:}\mathrm{=}\underset{\mathrm{\left(}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}}{\inf }\left\{\right.\underset{\mathit{j}\mathrm{\to }\mathrm{\infty }}{\lim \inf }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }{\mathit{u}}_{\mathit{j}}\mathrm{\right)}\mathrm{d}\mathit{x}\left|\begin{array}{c}\\ & \\ & \end{array}\right.\left\}\right.\mathit{,}& & \end{array}$

\hbox{$r\in[1,\frac{n}{n-1})$}$\mathit{r}\mathrm{\in }\mathrm{\left[}\mathrm{1}\mathit{,}\frac{\mathit{n}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

\begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega}f(\nabla u (x))\ud x + \int_{\Omega}\finf\bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, \end{equation*}$F_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{\ge }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }\mathit{u}\mathrm{\right(}\mathit{x}\mathrm{\left)}\mathrm{\right)}\mathrm{d}\mathit{x}\mathrm{+}{\mathrm{\int }}_{\mathit{\Omega }}{\mathit{f}}^{\mathrm{\infty }}\left(\right.\frac{{\mathit{D}}^{\mathit{s}}\mathit{u}}{\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}}\left)\right.\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}\mathit{,}$

\hbox{$ f^{\infty}(\xi):= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}${\mathit{f}}^{\mathrm{\infty }}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{:}\mathrm{=}{\overline{\lim }}_{\mathit{t}\mathrm{\to }\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{t\xi }\mathrm{\right)}\mathit{/}\mathit{t}$

In this paper we propose and analyze a localized orthogonal decomposition (LOD) methodfor solving semi-linear elliptic problems with heterogeneous and highly variablecoefficient functions. This Galerkin-type method is based on a generalized finite elementbasis that spans a low dimensional multiscale space. The basis is assembled by performinglocalized linear fine-scale computations on small patches that have a diameter of orderH | log (H)| where H is the coarse mesh size. Without any assumptions onthe type of the oscillations in the coefficients, we give a rigorous proof for a linearconvergence of the H1-error with respect to the coarse meshsize even for rough coefficients. To solve the corresponding system of algebraicequations, we propose an algorithm that is based on a damped Newton scheme in themultiscale space.

Long time simulations of transport equations raise computational challenges since theyrequire both a large domain of calculation and sufficient accuracy. It is thereforeadvantageous, in terms of computational costs, to use a time varying adaptive mesh, withsmall cells in the region of interest and coarser cells where the solution is smooth.Biological models involving cell dynamics fall for instance within this framework and areoften non conservative to account for cell division. In that case the thresholdcontrolling the spatial adaptivity may have to be time-dependent in order to keep up withthe progression of the solution. In this article we tackle the difficulties arising whenapplying a Multiresolution method to a transport equation with discontinuous fluxesmodeling localized mitosis. The analysis of the numerical method is performed on asimplified model and numerical scheme. An original threshold strategy is proposed andvalidated thanks to extensive numerical tests. It is then applied to a biological model inboth cases of distributed and localized mitosis.

We derive asymptotic formulas for the solutions of the mixed boundary value problem forthe Poisson equation on the union of a thin cylindrical plate and several thin cylindricalrods. One of the ends of each rod is set into a hole in the plate and the other one issupplied with the Dirichlet condition. The Neumann conditions are imposed on the wholeremaining part of the boundary. Elements of the junction are assumed to have contrastingproperties so that the small parameter, i.e. the relative thickness,appears in the differential equation, too, while the asymptotic structures cruciallydepend on the contrastness ratio. Asymptotic error estimates are derived in anisotropicweighted Sobolev norms.

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of adiscontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuousone leads to still open modeling questions, as in particular the singularity of thesolution at the abrupt transition and the choice of the right unknown to formulate thescattering problem. To address these questions we propose a mathematical approach based onvariational formulations set in weighted Sobolev spaces. Considering the discontinuousimpedance as the limit of a continuous boundary condition, we prove that only the problemformulated in terms of the velocity potential converges to a well-posed problem. Moreoverwe identify the limit problem and determine some Kutta-like condition satisfied by thevelocity: its convective derivative must vanish at the ends of the impedance area. Finallywe justify why it is not possible to define limit problems for the pressure and thedisplacement. Numerical examples illustrate the convergence process.

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite generalpartial differential operators. The starting point of our analysis is the DPG methodintroduced by [Demkowicz et al., SIAM J. Numer. Anal.49 (2011) 1788–1809; Zitelli et al., J.Comput. Phys. 230 (2011) 2406–2432]. This discretization resultsin a sparse positive definite linear algebraic system which can be obtained from a saddlepoint problem by an element-wise Schur complement reduction applied to the test space.Here, we show that the abstract framework of saddle point problems and domaindecomposition techniques provide stability and a priori estimates. Toobtain efficient numerical algorithms, we use a second Schur complement reduction appliedto the trial space. This restricts the degrees of freedom to the skeleton. We construct apreconditioner for the skeleton problem, and the efficiency of the discretization and thesolution method is demonstrated by numerical examples.