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Pierre Ladevéze – One of the best experts on this subject based on the ideXlab platform.

  • Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics Problems
    Computational Mechanics, 2012
    Co-Authors: Julien Waeytens, Ludovic Chamoin, Pierre Ladevéze

    Abstract:

    A new method is developed to obtain guaranteed error bounds on pointwise quantities of interest for linear transient viscodynamics Problems. The calculation of strict error bounds is based on the concept of “constitutive relation error” (CRE) and the solution of an Adjoint Problem. The central and original point of this work is the treatment of the singularity in space and time introduced by the loading of the Adjoint Problem. Hence, the Adjoint solution is decomposed into two parts: (i) an analytical part determined from Green’s functions; (ii) a residual part approximated with classical numerical tools (finite element method, Newmark integration scheme). The capabilities and the limits of the proposed approach are analyzed on a 2D example.

  • Strict and practical bounds through a non-intrusive and goal-oriented error estimation method for linear viscoelasticity Problems
    Finite Elements in Analysis and Design, 2009
    Co-Authors: Ludovic Chamoin, Pierre Ladevéze

    Abstract:

    In this work, we set up a non-intrusive procedure that yields for strict and high-quality error bounds of quantities of interest in linear viscoelasticity Problems solved by means of the finite element method (FEM). The goal-oriented error estimation approach uses the concept of dissipation error and classical duality techniques involving the solution of an Adjoint Problem. The non-intrusive feature of this approach is achieved by introducing enrichment functions, via a partition of unity, when solving the Adjoint Problem numerically (handbook techniques), so that the discretization parameters defined for the primal Problem can be reused. The resulting local error estimation method is thus highly effective, easy to implement in a finite element code, and it enables to consider discretization error on truly pointwise quantities of interest.

  • A non-intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity Problems
    Computer Methods in Applied Mechanics and Engineering, 2008
    Co-Authors: Ludovic Chamoin, Pierre Ladevéze

    Abstract:

    This paper presents a non-intrusive method which yields strict and high-quality error bounds of calculated quantities of interest in structural Problems solved by the finite element method. The focus is on linear viscoelasticity Problems described through internal variables. In order to solve the Adjoint Problem in a non-intrusive way, we use handbook techniques involving enrichment functions introduced through the partition of unity method (PUM). Then, the mesh and operators used to calculate the reference Problem can be reused. Such a procedure also enables one to address error estimation on pointwise quantities of interest, although this implies dealing with infinite-energy Green functions.

Gongsheng Li – One of the best experts on this subject based on the ideXlab platform.

  • conditional well posedness for an inverse source Problem in the diffusion equation using the variational Adjoint method
    Advances in Mathematical Physics, 2017
    Co-Authors: Gongsheng Li

    Abstract:

    This article deals with an inverse Problem of determining a linear source term in the multidimensional diffusion equation using the variational Adjoint method. A variational identity connecting the known data with the unknown is established based on an Adjoint Problem, and a conditional uniqueness for the inverse source Problem is proved by the approximate controllability to the Adjoint Problem under the condition that the unknowns can keep orders locally. Furthermore, a bilinear form is set forth also based on the variational identity and then a norm for the unknowns is well-defined by which a conditional Lipschitz stability is established.

  • conditional stability for an inverse Problem of determining a space dependent source coefficient in the advection dispersion equation with robin s boundary condition
    Abstract and Applied Analysis, 2014
    Co-Authors: Shunqin Wang, Gongsheng Li

    Abstract:

    This paper deals with an inverse Problem of determining the space-dependent source coefficient in one-dimensional advection-dispersion equation with Robin’s boundary condition. Data compatibility for the inverse Problem is analyzed by which an admissible set for the
    unknown is set forth. Furthermore, with the help of an integral identity, a conditional Lipschitz stability is established by suitably controlling the solution of an Adjoint Problem.

Vitali Vougalter – One of the best experts on this subject based on the ideXlab platform.

  • On the solvability conditions for the diffusion equation with convection terms.
    Communications on Pure and Applied Mathematics, 2012
    Co-Authors: Vitaly Volpert, Vitali Vougalter

    Abstract:

    Linear second order elliptic equation describing heat or mass di®usion and convection on a given velocity ¯eld is considered in R3. The corresponding operator L may not satisfy the Fredholm property. In this case, solvability conditions for the equation Lu = f are not known. In this work, we derive solvability conditions in H2(R3) for the non self-Adjoint Problem by relating it to a self-Adjoint SchrÄodinger type operator, for which solvability con- ditions are obtained in our previous work