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Unconditional convergence of DIRK schemes applied to dissipative evolution equations

Hansen, Eskil LU and Ostermann, Alexander (2010) In Applied Numerical Mathematics 60(1-2). p.55-63
Abstract
In this paper we prove the convergence of algebraically stable DIRK schemes applied to dissipative evolution equations on Hilbert spaces. The convergence analysis is unconditional as we do not impose any restrictions on the initial value or assume any extra regularity of the solution. The analysis is based on the observation that the schemes are linear combinations of the Yosida approximation, which enables the usage of an abstract approximation result for dissipative maps. The analysis is also extended to the case where the dissipative vector field is perturbed by a locally Lipschitz continuous map. The efficiency and robustness of these schemes are finally illustrated by applying them to a nonlinear diffusion equation.
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Dissipative evolution equations, DIRK schemes, Convergence, Nonlinear parabolic problems
in
Applied Numerical Mathematics
volume
60
issue
1-2
pages
55 - 63
publisher
Elsevier
external identifiers
  • wos:000272696600005
  • scopus:71549166659
ISSN
0168-9274
DOI
10.1016/j.apnum.2009.09.001
language
English
LU publication?
yes
id
e346bf34-f984-4a46-b397-c3db5d6f6256 (old id 1504009)
date added to LUP
2009-11-25 10:53:05
date last changed
2018-05-29 10:16:18
@article{e346bf34-f984-4a46-b397-c3db5d6f6256,
  abstract     = {In this paper we prove the convergence of algebraically stable DIRK schemes applied to dissipative evolution equations on Hilbert spaces. The convergence analysis is unconditional as we do not impose any restrictions on the initial value or assume any extra regularity of the solution. The analysis is based on the observation that the schemes are linear combinations of the Yosida approximation, which enables the usage of an abstract approximation result for dissipative maps. The analysis is also extended to the case where the dissipative vector field is perturbed by a locally Lipschitz continuous map. The efficiency and robustness of these schemes are finally illustrated by applying them to a nonlinear diffusion equation.},
  author       = {Hansen, Eskil and Ostermann, Alexander},
  issn         = {0168-9274},
  keyword      = {Dissipative evolution equations,DIRK schemes,Convergence,Nonlinear parabolic problems},
  language     = {eng},
  number       = {1-2},
  pages        = {55--63},
  publisher    = {Elsevier},
  series       = {Applied Numerical Mathematics},
  title        = {Unconditional convergence of DIRK schemes applied to dissipative evolution equations},
  url          = {http://dx.doi.org/10.1016/j.apnum.2009.09.001},
  volume       = {60},
  year         = {2010},
}