The Experts below are selected from a list of 102 Experts worldwide ranked by ideXlab platform
Marco Picasso - One of the best experts on this subject based on the ideXlab platform.
-
An adaptive algorithm for the transport equation with time dependent velocity
SN Applied Sciences, 2020Co-Authors: Samuel Dubuis, Marco PicassoAbstract:An a posteriori error estimate is derived for the approximation of the transport equation with a time dependent transport velocity. Continuous, piecewise linear, anisotropic finite elements are used for space discretization, the Crank-Nicolson scheme scheme is proposed for time discretization. This paper is a generalization of Dubuis S, Picasso M (J Sci Comput 75(1):350–375, 2018) where the transport velocity was not depending on time. The a posteriori error estimate (upper bound) is shown to be sharp for anisotropic meshes, the involved constant being independent of the mesh aspect ratio. A quadratic reconstruction of the numerical solution is introduced in order to obtain an estimate that is order two in time. Error indicators corresponding to space and time are proposed, their accuracy is checked with non-adapted meshes and constant time steps. Then, an adaptive algorithm is introduced, allowing to adapt the meshes and time steps. Numerical experiments are presented when the exact solution has strong variations in space and time, illustrating the efficiency of the method. They indicate that the Effectivity Index is close to one and does not depend on the solution, mesh size, aspect ratio, and time step.
-
An adaptive finite element method for the wave equation based on anisotropic a posteriori error estimates in the L2(H1) norm
2009Co-Authors: Marco PicassoAbstract:An adaptive finite element algorithm is presented for the wave equation in two space dimensions. The goal of the adaptive algorithm is to control the error in the same norm as for parabolic problems, namely the L2(0,T;H1(\Omega)) norm, where T denotes the final time and Omega the computational domain. The mesh aspect ratio can be large whenever needed, thus allowing a given level of accuracy to be reached with fewer vertices than with classical isotropic meshes. The refinement and coarsening criteria are based on anisotropic, a posteriori error estimates and on an elliptic reconstruction. A numerical study of the Effectivity Index on non-adapted meshes confirms the sharpness of the error estimator. Numerical results on adapted meshes indicate that the error indicator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error indicator becomes sharp again when considering the damped wave equation with a large damping coefficient, thus when the parabolic character of the PDE becomes predominant.
-
adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives
Computer Methods in Applied Mechanics and Engineering, 2006Co-Authors: Marco PicassoAbstract:An anisotropic error estimator involving only first order derivatives is proposed for the Laplace problem and continuous, piecewise linear finite elements. Upper and lower bounds are presented, the involved constants being independent of the mesh aspect ratio provided the error gradient is equidistributed in the directions of maximum and minimum stretching. An anisotropic adaptive algorithm is then proposed, with aim to equidistribute the error gradient in the directions of maximum and minimum stretching. Numerical results in two and three space dimensions show that the Effectivity Index is aspect ratio independent on such adapted meshes.
-
a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification
Computer Methods in Applied Mechanics and Engineering, 2003Co-Authors: O Kruger, Marco Picasso, Jeanfrancois ScheidAbstract:A posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L-2 in time H-1 in space error is bounded above and below by an error estimator based on the equation residual. Numerical results show that the Effectivity Index is close to one. An adaptive finite element algorithm is proposed and a solutal. dendrite is computed. (C) 2002 Elsevier Science B.V. All rights reserved.
-
numerical study of the Effectivity Index for an anisotropic error indicator based on zienkiewicz zhu error estimator
Communications in Numerical Methods in Engineering, 2002Co-Authors: Marco PicassoAbstract:The framework of Formaggia and Perotto (Numerische Mathematik 2001; 89:641-667) is considered to derive a new anisotropic error indicator for a Laplace problem in the energy norm. The matrix containing the error gradient is approached using a Zienkiewicz-Zhu error estimator. A numerical study of the Effectivity Index is proposed for anisotropic unstructured meshes, showing that our indicator is sharp. An anisotropic adaptive algorithm is implemented, aiming at controlling the estimated relative error. Copyright (C) 2003 John Wiley Sons, Ltd.
Jeanfrancois Scheid - One of the best experts on this subject based on the ideXlab platform.
-
a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification
Computer Methods in Applied Mechanics and Engineering, 2003Co-Authors: O Kruger, Marco Picasso, Jeanfrancois ScheidAbstract:A posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L-2 in time H-1 in space error is bounded above and below by an error estimator based on the equation residual. Numerical results show that the Effectivity Index is close to one. An adaptive finite element algorithm is proposed and a solutal. dendrite is computed. (C) 2002 Elsevier Science B.V. All rights reserved.
O Kruger - One of the best experts on this subject based on the ideXlab platform.
-
a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification
Computer Methods in Applied Mechanics and Engineering, 2003Co-Authors: O Kruger, Marco Picasso, Jeanfrancois ScheidAbstract:A posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L-2 in time H-1 in space error is bounded above and below by an error estimator based on the equation residual. Numerical results show that the Effectivity Index is close to one. An adaptive finite element algorithm is proposed and a solutal. dendrite is computed. (C) 2002 Elsevier Science B.V. All rights reserved.
Xu Wen - One of the best experts on this subject based on the ideXlab platform.
-
Relevant numerical methods for meso-scale wave propagation in heterogeneous media
2018Co-Authors: Xu WenAbstract:Les travaux de la présente thèse portent sur l’estimation d'erreur a posteriori pour les solutions numériques par éléments finis de l'équation des ondes élastiques dans les milieux hétérogènes. Deux types d’estimation ont été développés. Le premier considère directement l’équation élastodynamique et conduit à un nouvel estimateur d'erreur a posteriori explicite en norme L∞ en temps. Les principales caractéristiques de cet estimateur explicite sont l'utilisation de la méthode de résidus et le développement de reconstructions en temps et en espace selon les différentes régularités exigées par les différents termes contribuant à l’obtention d’une borne supérieure. L’analyse numérique de cet estimateur dans le cas des maillages uniformes montre qu’il assure bien une borne supérieure mais avec une propriété asymptotique qui reste à améliorer. Le deuxième type d’estimateur d’erreur est développé dans le contexte de la propagation des ondes à haute fréquence dans des milieux hétérogènes à l’échelle mésoscopique. Il s’agit d’une nouvelle erreur en résidus basée sur l'équation de transfert radiatif, qui est obtenue par un développement asymptotique multi-échelle de l'équation d'onde en utilisant la transformation de Wigner en espace-temps. Les résidus sont exprimés en termes de densités énergétiques calculés dans l’espace des phases pour les solutions d’onde numériques transitoires par éléments finis. L’analyse numérique de cette erreur appliquée aux milieux homogènes et hétérogènes en 1D a permis de valider notre approche. Les champs d’application visés sont la propagation des ondes sismiques dans les milieux géophysiques ou la propagation des ondes ultrasonores dans les milieux polycristallins.This thesis work deals with a posteriori error estimates for finite element solutions of the elastic wave equation in heterogeneous media. Two different a posteriori estimation approaches are developed. The first one, in a classical way, considers directly the elastodynamic equation and results in a new explicit error estimator in a non-natural L∞ norm in time. Its key features are the use of the residual method and the development of space and time reconstructions with respect to regularities required by different residual operators contributing to the proposed error bound. Numerical applications of the error bound with different mesh sizes show that it gives rise to a fully computable upper bound. However, its Effectivity Index and its asymptotic accuracy remain to be improved. The second error estimator is derived for high frequency wave propagation problem in heterogeneous media in the weak coupling regime. It is a new residual-type error based on the radiative transfer equation, which is derived by a multi-scale asymptotic expansion of the wave equation in terms of the spatio-temporal Wigner transforms of wave fields. The residual errors are in terms of angularly resolved energy quantities of numerical solutions of waves by finite element method. Numerical calculations of the defined errors in 1D homogeneous and heterogeneous media allow validating the proposed error estimation approach. The application field of this work is the numerical modelling of the seismic wave propagation in geophysical media or the ultrasonic wave propagation in polycrystalline materials
-
Méthodes d’analyse et de modélisation pertinentes pour la propagation des ondes à l’échelle méso dans des milieux hétérogènes
HAL CCSD, 2018Co-Authors: Xu WenAbstract:This thesis work deals with a posteriori error estimates for finite element solutions of the elastic wave equation in heterogeneous media. Two different a posteriori estimation approaches are developed. The first one, in a classical way, considers directly the elastodynamic equation and results in a new explicit error estimator in a non-natural L∞ norm in time. Its key features are the use of the residual method and the development of space and time reconstructions with respect to regularities required by different residual operators contributing to the proposed error bound. Numerical applications of the error bound with different mesh sizes show that it gives rise to a fully computable upper bound. However, its Effectivity Index and its asymptotic accuracy remain to be improved. The second error estimator is derived for high frequency wave propagation problem in heterogeneous media in the weak coupling regime. It is a new residual-type error based on the radiative transfer equation, which is derived by a multi-scale asymptotic expansion of the wave equation in terms of the spatio-temporal Wigner transforms of wave fields. The residual errors are in terms of angularly resolved energy quantities of numerical solutions of waves by finite element method. Numerical calculations of the defined errors in 1D homogeneous and heterogeneous media allow validating the proposed error estimation approach. The application field of this work is the numerical modelling of the seismic wave propagation in geophysical media or the ultrasonic wave propagation in polycrystalline materials.Les travaux de la présente thèse portent sur l’estimation d'erreur a posteriori pour les solutions numériques par éléments finis de l'équation des ondes élastiques dans les milieux hétérogènes. Deux types d’estimation ont été développés. Le premier considère directement l’équation élastodynamique et conduit à un nouvel estimateur d'erreur a posteriori explicite en norme L∞ en temps. Les principales caractéristiques de cet estimateur explicite sont l'utilisation de la méthode de résidus et le développement de reconstructions en temps et en espace selon les différentes régularités exigées par les différents termes contribuant à l’obtention d’une borne supérieure. L’analyse numérique de cet estimateur dans le cas des maillages uniformes montre qu’il assure bien une borne supérieure mais avec une propriété asymptotique qui reste à améliorer. Le deuxième type d’estimateur d’erreur est développé dans le contexte de la propagation des ondes à haute fréquence dans des milieux hétérogènes à l’échelle mésoscopique. Il s’agit d’une nouvelle erreur en résidus basée sur l'équation de transfert radiatif, qui est obtenue par un développement asymptotique multi-échelle de l'équation d'onde en utilisant la transformation de Wigner en espace-temps. Les résidus sont exprimés en termes de densités énergétiques calculés dans l’espace des phases pour les solutions d’onde numériques transitoires par éléments finis. L’analyse numérique de cette erreur appliquée aux milieux homogènes et hétérogènes en 1D a permis de valider notre approche. Les champs d’application visés sont la propagation des ondes sismiques dans les milieux géophysiques ou la propagation des ondes ultrasonores dans les milieux polycristallins
Mahboub Baccouch - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems
Journal of Scientific Computing, 2018Co-Authors: Mahboub BaccouchAbstract:In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the $$L^2$$ L 2 -norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve $$p+1$$ p + 1 order of convergence for the solution and its spatial derivative in the $$L^2$$ L 2 -norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order $$p+1$$ p + 1 towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order $$p+3/2$$ p + 3 / 2 towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the $$(p+1)$$ ( p + 1 ) -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the $$L^2$$ L 2 -norm at $$\mathcal {O}(h^{p+3/2})$$ O ( h p + 3 / 2 ) rate. Finally, we prove that the global Effectivity Index in the $$L^2$$ L 2 -norm converge to unity at $$\mathcal {O}(h^{1/2})$$ O ( h 1 / 2 ) rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$ P p polynomials with $$p\ge 1$$ p ≥ 1 . Finally, several numerical examples are given to validate the theoretical results.
-
Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension
Journal of Scientific Computing, 2016Co-Authors: Mahboub BaccouchAbstract:In this paper, we propose and analyze a robust recovery-based error estimator for the original discontinuous Galerkin method for nonlinear scalar conservation laws in one space dimension. The proposed a posteriori error estimator of the recovery-type is easy to implement, computationally simple, asymptotically exact, and is useful in adaptive computations. We use recent results (Meng et al. in SIAM J Numer Anal 50:2336–2356, 2012 ) to prove that, for smooth solutions, our a posteriori error estimates at a fixed time converge to the true spatial errors in the $$L^2$$ L 2 -norm under mesh refinement. The order of convergence is proved to be $$p + 1$$ p + 1 , when $$p$$ p -degree piecewise polynomials with $$p\ge 1$$ p ≥ 1 are used. We further prove that the global Effectivity Index converges to unity at $$\mathcal {O}(h)$$ O ( h ) rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$ P p polynomials with $$p\ge 1$$ p ≥ 1 , under the condition that $$|f'(u)|$$ | f ′ ( u ) | possesses a uniform positive lower bound, where $$f(u)$$ f ( u ) is the nonlinear flux function. We provide several numerical examples to support our theoretical results, to show the effectiveness of our recovery-based a posteriori error estimates, and to demonstrate that our results hold true for nonlinear conservation laws with general flux functions. These experiments indicate that the restriction on $$f(u)$$ f ( u ) is artificial.
