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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...

Galerkin approximations for the linear parabolic equation with the third boundary condition

István Faragó, Sergey Korotov, Pekka Neittaanmäki (2003)

Applications of Mathematics

We solve a linear parabolic equation in ℝ d , d ≥ 1 , with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the θ -method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.

Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems

Martin Kahlbacher, Stefan Volkwein (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functions and on the parameter grid that is used to generate the snapshots and to compute the POD basis. The...

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2004)

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

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

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