Additive manufacturing by laser fusion on metal oxides powder bed such as e.g. alumina (Al2O3) or... more Additive manufacturing by laser fusion on metal oxides powder bed such as e.g. alumina (Al2O3) or aluminium titanate (Al2T iO5) has developed considerably in the last few years and allows today the production of a wide range of complex objects. The mathematical problem considered is to control the temperature inside some part Ω of a powder layer. This phenomenon is governed by a parabolic initial boundary value problem with a heat source corresponding to the laser trajectory on some part of the boundary ∂Ω. The main questions concern the optimization of the trajectories scanned by the laser on the boundary ∂Ω according to given criteria: imposing that during the thermal process the temperature reaches a melting value in the structure to be built, a desired temperature distribution at the end of the thermal process, minimizing the thermal gradients, all this in the shortest possible thermal treatment time. To achieve this goal, we start by proving the existence of an optimal control, followed by first order necessary optimality conditions. Finally, we establish a second order sufficient optimality condition. Keywords. Optimal control problems, parabolic equations, heat equations with moving source, trajectories, time of thermal treatment, cost functionals, existence of an optimal control, adjoint problem, first order necessary optimality conditions, second order sufficient optimality conditions.
Mathematical Methods in The Applied Sciences, Jul 18, 2016
We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhe... more We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face, and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the Finite Element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis.
Mediterranean Journal of Mathematics, Oct 29, 2022
In this paper, we investigate the direct and indirect stability of locally coupled wave equations... more In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate t − 1 2 if the two waves have the same speed of propagation, and with rate t − 1 3 if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order t −1 .
Numerical Functional Analysis and Optimization, Nov 23, 2006
This Note presents an a posteriori error estimator of residual type for the stationary Stokes pro... more This Note presents an a posteriori error estimator of residual type for the stationary Stokes problem using the dual mixed FEM. We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution. To cite this article: M.
We present new a posteriori error estimates for the finite volume approximations of elliptic prob... more We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires solving only finite-dimensional problems.
We introduce a new H(div) flux reconstruction for discontinuous Galerkin approximations of ellipt... more We introduce a new H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. The reconstructed flux is computed elementwise and its divergence equals the L 2-orthogonal projection of the source term onto the discrete space. Moreover, the energy-norm of the error in the flux is bounded by the discrete energy-norm of the error in the primal variable, independently of diffusion heterogeneities. To cite this article: A.
A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient ... more A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes.
In this paper, a general form of functional type a posteriori error estimates for linear reaction... more In this paper, a general form of functional type a posteriori error estimates for linear reaction-convection-diffusion problems is presented. It is derived by purely functional arguments without attracting specific properties of the approximation method. The estimate provides a guaranteed upper bound of the difference between the exact solution and any conforming approximation from the energy functional class. It is also proved that the derived error majorants give computable quantities, which are equivalent to the error evaluated in the energy and combined primal-dual norms. Bibliography: 14 titles.
The very weak solution of the Poisson equation with L 2 boundary data is defined by the method of... more The very weak solution of the Poisson equation with L 2 boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the L 2 (Ω)-norm with order 1/2 in convex domains but has a reduced convergence order in non-convex domains although the solution remains to be contained in H 1/2 (Ω). The reason is a singularity in the dual problem. In this paper we propose and analyze, as a remedy, both a standard finite element method with mesh grading and a dual variant of the singular complement method. The error order 1/2 is retained in both cases also with non-convex domains. Numerical experiments confirm the theoretical results.
HAL (Le Centre pour la Communication Scientifique Directe), Jan 24, 2023
In this work, we propose an a posteriori goal-oriented error estimator for the harmonic A-ϕ formu... more In this work, we propose an a posteriori goal-oriented error estimator for the harmonic A-ϕ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Our theoretical results are illustrated by numerical experiments.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 8, 2018
In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations ... more In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains similar to the one for domains with smooth boundary proposed in [14, §4.5.d]. It turns out that the variational space is embedded in H 1 as soon as the domain satisfies a certain geometrical assumption (in particular it holds for convex polyhedra). In such a case, the existence of a weak solution follows by a compact perturbation argument. As the associated boundary value problem is an elliptic system, standard shift theorem from [16] can be applied if the corner and edge singularities are explicitly known. We therefore describe such singularities, by adapting the general strategy from [13]. Finally in order to perform a wavenumber explicit error analysis of our problem, a stability estimate is mandatory (see [32, 33] for the Helmholtz equation). We then prove such an estimate for some particular configurations. We end up with the study of a Galerkin (h-version) finite element method using Lagrange elements and give wave number explicit error bounds in the asymptotic ranges. Some numerical tests that illustrate our theoretical results are also presented.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 12, 2013
The A/ϕ magnetodynamic Maxwell system given in its potential and space/time formulation is a popu... more The A/ϕ magnetodynamic Maxwell system given in its potential and space/time formulation is a popular model considered in the engineering community. We establish exitence of strong solutions with the help of the theory of Showalter on degenerated parabolic problems; using energy estimates, existence of weak solutions are also deduced.
We consider the variational formulation of the electric field integral equation on a Lipschitz po... more We consider the variational formulation of the electric field integral equation on a Lipschitz polyhedral surface Γ. We study the Galerkin boundary element discretisations based on the lowest-order Raviart-Thomas surface elements on a sequence of anisotropic meshes algebraically graded towards the edges of Γ. We establish quasioptimal convergence of Galerkin solutions under a mild restriction on the strength of grading. The key ingredient of our convergence analysis are new componentwise stability properties of the Raviart-Thomas interpolant on anisotropic elements.
We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary ... more We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 1, 2013
We show that the solution of the two-dimensional Dirichlet problem set in a plane domain is the l... more We show that the solution of the two-dimensional Dirichlet problem set in a plane domain is the limit of the solutions of similar problems set on a sequence of one-dimensional networks as their size goes to zero. Roughly speaking this means that a membrane can be seen as the limit of rackets made of strings. For practical applications, we also show that the solutions of the discrete approximated problems (again on the one-dimensional networks) also converge to the solution of the two-dimensional Dirichlet problem.
Several approaches are discussed how to understand the solution of the Dirichlet problem for the ... more Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in L 2 only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi-uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to 2π.
HAL (Le Centre pour la Communication Scientifique Directe), 2017
In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodyna... more In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodynamic formulation of the Maxwell's system. This system is recast in two classical potential formulations, which are solved by a Finite Element method. The equilibrated estimator is built starting from these dual problems and is consequently available to estimate the error of both numerical resolutions. The estimator reliability and efficiency without generic constants are established. Then, numerical tests are performed, allowing to illustrate the obtained theoretical results.
Journal of Dynamics and Differential Equations, Jul 30, 2021
In this paper we analyze a coupled system between a transport equation and an ordinary differenti... more In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable. Also, we perform a bifurcation analysis and determine some properties of the stable steady state set and the limit cycle oscillation region. Some numerical examples illustrate the theoretical results.
Additive manufacturing by laser fusion on metal oxides powder bed such as e.g. alumina (Al2O3) or... more Additive manufacturing by laser fusion on metal oxides powder bed such as e.g. alumina (Al2O3) or aluminium titanate (Al2T iO5) has developed considerably in the last few years and allows today the production of a wide range of complex objects. The mathematical problem considered is to control the temperature inside some part Ω of a powder layer. This phenomenon is governed by a parabolic initial boundary value problem with a heat source corresponding to the laser trajectory on some part of the boundary ∂Ω. The main questions concern the optimization of the trajectories scanned by the laser on the boundary ∂Ω according to given criteria: imposing that during the thermal process the temperature reaches a melting value in the structure to be built, a desired temperature distribution at the end of the thermal process, minimizing the thermal gradients, all this in the shortest possible thermal treatment time. To achieve this goal, we start by proving the existence of an optimal control, followed by first order necessary optimality conditions. Finally, we establish a second order sufficient optimality condition. Keywords. Optimal control problems, parabolic equations, heat equations with moving source, trajectories, time of thermal treatment, cost functionals, existence of an optimal control, adjoint problem, first order necessary optimality conditions, second order sufficient optimality conditions.
Mathematical Methods in The Applied Sciences, Jul 18, 2016
We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhe... more We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face, and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the Finite Element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis.
Mediterranean Journal of Mathematics, Oct 29, 2022
In this paper, we investigate the direct and indirect stability of locally coupled wave equations... more In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate t − 1 2 if the two waves have the same speed of propagation, and with rate t − 1 3 if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order t −1 .
Numerical Functional Analysis and Optimization, Nov 23, 2006
This Note presents an a posteriori error estimator of residual type for the stationary Stokes pro... more This Note presents an a posteriori error estimator of residual type for the stationary Stokes problem using the dual mixed FEM. We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution. To cite this article: M.
We present new a posteriori error estimates for the finite volume approximations of elliptic prob... more We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires solving only finite-dimensional problems.
We introduce a new H(div) flux reconstruction for discontinuous Galerkin approximations of ellipt... more We introduce a new H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. The reconstructed flux is computed elementwise and its divergence equals the L 2-orthogonal projection of the source term onto the discrete space. Moreover, the energy-norm of the error in the flux is bounded by the discrete energy-norm of the error in the primal variable, independently of diffusion heterogeneities. To cite this article: A.
A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient ... more A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes.
In this paper, a general form of functional type a posteriori error estimates for linear reaction... more In this paper, a general form of functional type a posteriori error estimates for linear reaction-convection-diffusion problems is presented. It is derived by purely functional arguments without attracting specific properties of the approximation method. The estimate provides a guaranteed upper bound of the difference between the exact solution and any conforming approximation from the energy functional class. It is also proved that the derived error majorants give computable quantities, which are equivalent to the error evaluated in the energy and combined primal-dual norms. Bibliography: 14 titles.
The very weak solution of the Poisson equation with L 2 boundary data is defined by the method of... more The very weak solution of the Poisson equation with L 2 boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the L 2 (Ω)-norm with order 1/2 in convex domains but has a reduced convergence order in non-convex domains although the solution remains to be contained in H 1/2 (Ω). The reason is a singularity in the dual problem. In this paper we propose and analyze, as a remedy, both a standard finite element method with mesh grading and a dual variant of the singular complement method. The error order 1/2 is retained in both cases also with non-convex domains. Numerical experiments confirm the theoretical results.
HAL (Le Centre pour la Communication Scientifique Directe), Jan 24, 2023
In this work, we propose an a posteriori goal-oriented error estimator for the harmonic A-ϕ formu... more In this work, we propose an a posteriori goal-oriented error estimator for the harmonic A-ϕ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Our theoretical results are illustrated by numerical experiments.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 8, 2018
In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations ... more In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains similar to the one for domains with smooth boundary proposed in [14, §4.5.d]. It turns out that the variational space is embedded in H 1 as soon as the domain satisfies a certain geometrical assumption (in particular it holds for convex polyhedra). In such a case, the existence of a weak solution follows by a compact perturbation argument. As the associated boundary value problem is an elliptic system, standard shift theorem from [16] can be applied if the corner and edge singularities are explicitly known. We therefore describe such singularities, by adapting the general strategy from [13]. Finally in order to perform a wavenumber explicit error analysis of our problem, a stability estimate is mandatory (see [32, 33] for the Helmholtz equation). We then prove such an estimate for some particular configurations. We end up with the study of a Galerkin (h-version) finite element method using Lagrange elements and give wave number explicit error bounds in the asymptotic ranges. Some numerical tests that illustrate our theoretical results are also presented.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 12, 2013
The A/ϕ magnetodynamic Maxwell system given in its potential and space/time formulation is a popu... more The A/ϕ magnetodynamic Maxwell system given in its potential and space/time formulation is a popular model considered in the engineering community. We establish exitence of strong solutions with the help of the theory of Showalter on degenerated parabolic problems; using energy estimates, existence of weak solutions are also deduced.
We consider the variational formulation of the electric field integral equation on a Lipschitz po... more We consider the variational formulation of the electric field integral equation on a Lipschitz polyhedral surface Γ. We study the Galerkin boundary element discretisations based on the lowest-order Raviart-Thomas surface elements on a sequence of anisotropic meshes algebraically graded towards the edges of Γ. We establish quasioptimal convergence of Galerkin solutions under a mild restriction on the strength of grading. The key ingredient of our convergence analysis are new componentwise stability properties of the Raviart-Thomas interpolant on anisotropic elements.
We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary ... more We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 1, 2013
We show that the solution of the two-dimensional Dirichlet problem set in a plane domain is the l... more We show that the solution of the two-dimensional Dirichlet problem set in a plane domain is the limit of the solutions of similar problems set on a sequence of one-dimensional networks as their size goes to zero. Roughly speaking this means that a membrane can be seen as the limit of rackets made of strings. For practical applications, we also show that the solutions of the discrete approximated problems (again on the one-dimensional networks) also converge to the solution of the two-dimensional Dirichlet problem.
Several approaches are discussed how to understand the solution of the Dirichlet problem for the ... more Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in L 2 only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi-uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to 2π.
HAL (Le Centre pour la Communication Scientifique Directe), 2017
In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodyna... more In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodynamic formulation of the Maxwell's system. This system is recast in two classical potential formulations, which are solved by a Finite Element method. The equilibrated estimator is built starting from these dual problems and is consequently available to estimate the error of both numerical resolutions. The estimator reliability and efficiency without generic constants are established. Then, numerical tests are performed, allowing to illustrate the obtained theoretical results.
Journal of Dynamics and Differential Equations, Jul 30, 2021
In this paper we analyze a coupled system between a transport equation and an ordinary differenti... more In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable. Also, we perform a bifurcation analysis and determine some properties of the stable steady state set and the limit cycle oscillation region. Some numerical examples illustrate the theoretical results.
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