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A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – Analysis, assessments and applications to parameter estimation

D. Chapelle, A. Gariah, P. Moireau, J. Sainte-Marie (2013)

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

We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the...

A gradient estimate for solutions of the heat equation. II

Charles S. Kahane (2001)

Czechoslovak Mathematical Journal

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.

A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

David J. Knezevic, Endre Süli (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker–Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier–Stokes–Fokker–Planck system for dilute polymeric fluids. In this context the Fokker–Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for...

A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method

Abdallah Bradji, Jürgen Fuhrmann (2014)

Mathematica Bohemica

Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of 𝒲 1 , ( 2 ) is proved. An ( 1 ) -error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...

A note on nonhomogeneous initial and boundary conditions in parabolic problems solved by the Rothe method

Karel Rektorys, Marie Ludvíková (1980)

Aplikace matematiky

When solving parabolic problems by the so-called Rothe method (see K. Rektorys, Czech. Math. J. 21 (96), 1971, 318-330 and other authors), some difficulties of theoretical nature are encountered in the case of nonhomogeneous initial and boundary conditions. As a rule, these difficulties lead to rather unnatural additional conditions imposed on the corresponding bilinear form and the initial and boundary functions. In the present paper, it is shown how to remove such additional assumptions in the...

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