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 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 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.

This paper deals with the numerical study of a nonlinear, strongly anisotropic heatequation. The use of standard schemes in this situation leads to poor results, due to thehigh anisotropy. An Asymptotic-Preserving method is introduced in this paper, which issecond-order accurate in both, temporal and spacial variables. The discretization in timeis done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to beindependent of the anisotropy parameter , andthis for fixed coarse Cartesian grids and for variable anisotropy directions. The contextof this work are magnetically confined fusion plasmas.

For scalar conservation laws in one space dimension with a flux function discontinuous inspace, there exist infinitely many classes of solutions which are L1 contractive.Each class is characterized by a connection (A,B) which determines the interface entropy. Forsolutions corresponding to a connection (A,B), there exists convergent numerical schemesbased on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes,corresponding to computationally less expensive monotone schemes like Lax−Friedrichs etc., usedwidely in applications. In this paper we completely answer this question for more general(A,B)stable monotone schemes using a novel construction of interface flux function. Then fromthe singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, weprove the convergence of the schemes.

The chemical master equation is a fundamental equation in chemical kinetics. It underliesthe classical reaction-rate equations and takes stochastic effects into account. In thispaper we give a simple argument showing that the solutions of a large class of chemicalmaster equations are bounded in weighted ℓ1-spaces and possess high-ordermoments. This class includes all equations in which no reactions between two or morealready present molecules and further external reactants occur that add mass to thesystem. As an illustration for the implications of this kind of regularity, we analyze theeffect of truncating the state space. This leads to an error analysis for the finite stateprojections of the chemical master equation, an approximation that forms the basis of manynumerical methods.

In this paper, we propose a method for the approximation of the solution ofhigh-dimensional weakly coercive problems formulated in tensor spaces using low-rankapproximation formats. The method can be seen as a perturbation of a minimal residualmethod with a measure of the residual corresponding to the error in a specified solutionnorm. The residual norm can be designed such that the resulting low-rank approximationsare optimal with respect to particular norms of interest, thus allowing to take intoaccount a particular objective in the definition of reduced order approximations ofhigh-dimensional problems. We introduce and analyze an iterative algorithm that is able toprovide an approximation of the optimal approximation of the solution in a given low-ranksubset, without any a priori information on this solution. We alsointroduce a weak greedy algorithm which uses this perturbed minimal residual method forthe computation of successive greedy corrections in small tensor subsets. We prove itsconvergence under some conditions on the parameters of the algorithm. The proposednumerical method is applied to the solution of a stochastic partial differential equationwhich is discretized using standard Galerkin methods in tensor product spaces.

In this paper, we propose implicit and semi-implicit in time finite volume schemes forthe barotropic Euler equations (hence, as a particular case, for the shallow waterequations) and for the full Euler equations, based on staggered discretizations. Forstructured meshes, we use the MAC finite volume scheme, and, for general mixedquadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of theRannacher−Turek orCrouzeix−Raviart finiteelements. We first show that a solution to each of these schemes satisfies a discretekinetic energy equation. In the barotropic case, a solution also satisfies a discreteelastic potential balance; integrating these equations over the domain readily yieldsdiscrete counterparts of the stability estimates which are known for the continuousproblem. In the case of the full Euler equations, the scheme relies on the discretizationof the internal energy balance equation, which offers two main advantages: first, we avoidthe space discretization of the total energy, which involves cell-centered andface-centered variables; second, we obtain an algorithm which boils down to a usualpressure correction scheme in the incompressible limit. Consistency (in a weak sense) withthe original total energy conservative equation is obtained thanks to corrective terms inthe internal energy balance, designed to compensate numerical dissipation terms appearingin the discrete kinetic energy inequality. It is then shown in the 1D case, that,supposing the convergence of a sequence of solutions, the limit is an entropy weaksolution of the continuous problem in the barotropic case, and a weak solution in the fullEuler case. Finally, we present numerical results which confirm this theory.

This paper develops a framework to include Dirichlet boundary conditions on a subset ofthe boundary which depends on time. In this model, the boundary conditions are weaklyenforced with the help of a Lagrange multiplier method. In order to avoid that the ansatzspace of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, whichmaps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition aswell as existence results are presented for a class of second order initial-boundary valueproblems. For the semi-discretization in space, a finite element scheme is presented whichsatisfies a discrete stability condition. Because of the saddle point structure of theunderlying PDE, the resulting system is a DAE of index 3.