A note on space approximation of parabolic evolution equations
type: Item
instance of: scholarly article
title: A note on space approximation of parabolic evolution equations
zbMATH Open document ID: 1067.65057
author: Erika Hausenblas
published in: Applied Mathematics and Computation
publication date: 2004-10-14T00:00:00Z
review text: The problem is to find the numerical approximation of the abstract Cauchy problem \[ \dot u(t)= Au(t),\quad u(0)= x_0\in X, \] where \(A\) generates an analytic semigroup in a separable Banach space \(X\). Such equations include e.g. parabolic partial differential equations of arbitrary order defined on a bounded domain. The author investigates the accuracy of space approximation of solutions to parabolic evolution equations in arbitrary separable Banach space covering Besov spaces and \(L^p\) spaces, \(1< p< 2\). He formulates the problem in the abstract setting of semigroup theory and gives the rate of convergence depending on the smoothness of the initial condition in terms of real interpolation spaces. To handle regularity he introduces several scales of \(X\) which are available by interpolation methods.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 2107896
zbMATH Keywords: parabolic evolution equations
zbMATH Keywords: space approximation
zbMATH Keywords: abstract Cauchy problem
zbMATH Keywords: Banach space
zbMATH Keywords: Besov spaces
zbMATH Keywords: \(L^p\) spaces
zbMATH Keywords: semigroup theory
zbMATH Keywords: convergence
zbMATH Keywords: interpolation spaces
reviewed by: Otu Vaarmann
MaRDI profile type: Publication
full work available at URL: j.amc.2003.08.094
OpenAlex ID: W2020125648
Wikidata QID: Q59225727
cites work: Q4112938
cites work: Q5549343
cites work: Q4356573
cites work: Operator theory and numerical methods
cites work: Q4540535
cites work: Q3633377
cites work: Q4220655
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4360234
DBLP publication ID: journals/amc/Hausenblas04
DOI: 10.1016/J.AMC.2003.08.094
description: scientific article; zbMATH DE number 2107896