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Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out...

Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out...

Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out...

Geometric renormalization of large energy wave maps

Terence Tao (2004)

Journées Équations aux dérivées partielles

There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the...

Geometrical aspects of exact boundary controllability for the wave equation - a numerical study

M. Asch, G. Lebeau (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This essentially numerical study, sets out to investigate various geometrical properties of exact boundary controllability of the wave equation when the control is applied on a part of the boundary. Relationships between the geometry of the domain, the geometry of the controlled boundary, the time needed to control and the energy of the control are dealt with. A new norm of the control and an energetic cost factor are introduced. These quantities enable a detailed appraisal of the numerical solutions...

Homogenization of evolution problems for a composite medium with very small and heavy inclusions

Michel Bellieud (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We study the homogenization of parabolic or hyperbolic equations like ρ ε ∂ n u ε ∂ t n - div ( a ε ∇ u ε ) = f in Ω × ( 0 , T ) + boundary conditions , n ∈ { 1 , 2 } , when the coefficients ρ ε , a ε (defined in Ø ) take possibly high values on a ε -periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

Homogenization of evolution problems for a composite medium with very small and heavy inclusions

Michel Bellieud (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the homogenization of parabolic or hyperbolic equations like ρ ε ∂ n u ε ∂ t n - div ( a ε ∇ u ε ) = f in Ø × ( 0 , T ) + boundary conditions , n ∈ { 1 , 2 } , when the coefficients ρ ε , a ε (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian

Jamel El Kamel, Chokri Yacoub (2005)

Annales mathématiques Blaise Pascal

In this paper we consider the modified wave equation associated with a class of radial Laplacians L generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra c -function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of...

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