Coupling finite element and spectral methods: First results

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization is used on the second one. Two kinds of matching conditions on the interface are presented and compared. In both cases, error estimates are proved

Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data

International audienceThis paper is concerned with solving Cauchy problem for elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, Cauchy problem is presented as an optimal control problem. Numerical convergence analysis is carried out and leads to an adapted stopping criteria for the minimization process depending on noise rate. Numerical examples involving smooth and singular data are presented

A Finite Element Method for the Boundary Data Recovery in an Oxygen-Balance Dispersion Model

International audienceThe inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional transport model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the outfall point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequences of these non-standard boundary conditions is that dispersive transport equations on both oxygen tracers are strongly coupled and the resulting system becomes ill-posed. The main purpose is a finite element space-discretization of the variational problem put under a non-symmetric mixed form. Combining analytical calculations, numerical computations and theoretical justifications, we try to elucidate the characteristics related to the ill-posedness of this data completion dynamical problem and understand its mathematical structure

Numerical analysis of an energy-like minimization method to solve a parabolic Cauchy problem with noisy data

International audienceThis paper is concerned with solving Cauchy problem for parabolic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to an adapted stopping criteria depending on noise rate for the minimization process. Numerical experiments are performed and confirm theoretical convergence order and the good behavior of the minimization process

Adaptive anisotropic mesh generation : Theory and practical aspects

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Quasi-optimal triangulations for gradient nonconforming interpolates of piecewise regular functions

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Quasi-Optimal Meshes for Gradient Nonconforming Approximations

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Star-based a posteriori error estimates for elliptic problems.

International audienceWe give an a posteriori error estimator for nonconforming finite element approximations of diffusion-reaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior and efficiency of this estimator for generic elliptic problems

An alternative algorithm for regularization of noisy volatility calibration in Finance

International audienceThis contribution is an extension of the work initiated in [1], presenting a strategy for the calibration of the local volatility. Due to Morozov's discrepancy principle [6], the Tikhonov regularization problem introduced in [7] is understood as an inequality-constrained minimization problem. An Uzawa procedure is proposed to replace this latter by a sequence of unconstrained problems dealt with in the modified Thikonov regularization procedure in [1]. Numerical tests confirm the consistency of the approach and the significant speed-up of the process of local volatility determination.Cette contribution dans ce papier est une extension des travaux initiés dans [1], qui pré-sente une stratégie pour l'estimation de la volatilité locale. En raison du principe de la différence de Morozov [6], le problème de la régularisation de Tikhonov introduite dans [7] est reformulé comme un problème de minimisation de l'inégalité des contraintes. Une procédure Uzawa est proposé de remplacer ce dernier par une séquence de problèmes non contraints traités dans la procédure de régularisation Thikonov modifié dans [1]. Des tests numériques confirment la cohérence de l'approche et l'importante accélérer le processus de détermination de la volatilité locale