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In this article we develop a posteriori error estimates for second order
linear elliptic problems with point sources in two- and three-dimensional domains. We
prove a global upper bound and a local lower bound for the error measured in a weighted
Sobolev space. The weight considered is a (positive) power of the distance to the support
of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory
hinges on local approximation properties of either Clément or Scott–Zhang interpolation
operators, without need of modifications, and makes use of weighted estimates for
fractional integrals and maximal functions. Numerical experiments with an adaptive
algorithm 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 the
inclusions, we derive a coupled system of partial differential equations governing the
evolution of the temperature of each phase at a macroscopic level of description. The
coupling terms describing the exchange of heat between the phases are obtained by using
homogenization techniques originating from [D. Cioranescu, F. Murat, Collège de France
Seminar, 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 control
problems governed by parametrized parabolic partial differential equations. To this end,
we employ the reduced basis method as a low-dimensional surrogate model to solve the
optimal control problem and develop a posteriori error estimation
procedures that provide rigorous bounds for the error in the optimal control and the
associated cost functional. We show that our approach can be applied to problems involving
control constraints and that, even in the presence of control constraints, the reduced
order optimal control problem and the proposed bounds can be efficiently evaluated in an
offline-online computational procedure. We also propose two greedy sampling procedures to
construct the reduced basis space. Numerical results are presented to confirm the validity
of our approach.
In this paper, we are interested in modelling the flow of the coolant (water) in a
nuclear reactor core. To this end, we use a monodimensional low Mach number model
supplemented with the stiffened gas law. We take into account potential phase transitions
by a single equation of state which describes both pure and mixture phases. In some
particular cases, we give analytical steady and/or unsteady solutions which provide
qualitative information about the flow. In the second part of the paper, we introduce two
variants of a numerical scheme based on the method of characteristics to simulate this
model. We study and verify numerically the properties of these schemes. We finally present
numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip
We consider an initial-boundary value problem for a generalized 2D time-dependent
Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the
Crank–Nicolson-type finite-difference scheme with approximate or discrete transparent
boundary conditions (TBCs), the Strang-type splitting with respect to the potential is
applied. For the resulting method, the unconditional uniform in time L2-stability is
proved. Due to the splitting, an effective direct algorithm using FFT is developed now to
implement the method with the discrete TBC for general potential. Numerical results on the
tunnel effect for rectangular barriers are included together with the detailed practical
error analysis confirming nice properties of the method.
This paper deals with the numerical study of a nonlinear, strongly anisotropic heat
equation. The use of standard schemes in this situation leads to poor results, due to the
high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is
second-order accurate in both, temporal and spacial variables. The discretization in time
is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be
independent of the anisotropy parameter , and
this for fixed coarse Cartesian grids and for variable anisotropy directions. The context
of this work are magnetically confined fusion plasmas.
For scalar conservation laws in one space dimension with a flux function discontinuous in
space, 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. For
solutions corresponding to a connection (A,B), there exists convergent numerical schemes
based 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., used
widely in applications. In this paper we completely answer this question for more general
stable monotone schemes using a novel construction of interface flux function. Then from
the singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, we
prove the convergence of the schemes.
The chemical master equation is a fundamental equation in chemical kinetics. It underlies
the classical reaction-rate equations and takes stochastic effects into account. In this
paper we give a simple argument showing that the solutions of a large class of chemical
master equations are bounded in weighted ℓ1-spaces and possess high-order
moments. This class includes all equations in which no reactions between two or more
already present molecules and further external reactants occur that add mass to the
system. As an illustration for the implications of this kind of regularity, we analyze the
effect of truncating the state space. This leads to an error analysis for the finite state
projections of the chemical master equation, an approximation that forms the basis of many
In this paper, we propose a method for the approximation of the solution of
high-dimensional weakly coercive problems formulated in tensor spaces using low-rank
approximation formats. The method can be seen as a perturbation of a minimal residual
method with a measure of the residual corresponding to the error in a specified solution
norm. The residual norm can be designed such that the resulting low-rank approximations
are optimal with respect to particular norms of interest, thus allowing to take into
account a particular objective in the definition of reduced order approximations of
high-dimensional problems. We introduce and analyze an iterative algorithm that is able to
provide an approximation of the optimal approximation of the solution in a given low-rank
subset, without any a priori information on this solution. We also
introduce a weak greedy algorithm which uses this perturbed minimal residual method for
the computation of successive greedy corrections in small tensor subsets. We prove its
convergence under some conditions on the parameters of the algorithm. The proposed
numerical method is applied to the solution of a stochastic partial differential equation
which is discretized using standard Galerkin methods in tensor product spaces.
In this paper, we propose implicit and semi-implicit in time finite volume schemes for
the barotropic Euler equations (hence, as a particular case, for the shallow water
equations) and for the full Euler equations, based on staggered discretizations. For
structured meshes, we use the MAC finite volume scheme, and, for general mixed
quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the
elements. We first show that a solution to each of these schemes satisfies a discrete
kinetic energy equation. In the barotropic case, a solution also satisfies a discrete
elastic potential balance; integrating these equations over the domain readily yields
discrete counterparts of the stability estimates which are known for the continuous
problem. In the case of the full Euler equations, the scheme relies on the discretization
of the internal energy balance equation, which offers two main advantages: first, we avoid
the space discretization of the total energy, which involves cell-centered and
face-centered variables; second, we obtain an algorithm which boils down to a usual
pressure correction scheme in the incompressible limit. Consistency (in a weak sense) with
the original total energy conservative equation is obtained thanks to corrective terms in
the internal energy balance, designed to compensate numerical dissipation terms appearing
in 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 weak
solution of the continuous problem in the barotropic case, and a weak solution in the full
Euler case. Finally, we present numerical results which confirm this theory.
This paper develops a framework to include Dirichlet boundary conditions on a subset of
the boundary which depends on time. In this model, the boundary conditions are weakly
enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz
space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which
maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as
well as existence results are presented for a class of second order initial-boundary value
problems. For the semi-discretization in space, a finite element scheme is presented which
satisfies a discrete stability condition. Because of the saddle point structure of the
underlying PDE, the resulting system is a DAE of index 3.