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  4. An Anisotropic Error Estimator For The Crank-Nicolson Method: Application To A Parabolic Problem
 
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research article

An Anisotropic Error Estimator For The Crank-Nicolson Method: Application To A Parabolic Problem

Lozinski, Alexei  
•
Picasso, Marco  
•
Prachittham, Virabouth
2009
Siam Journal On Scientific Computing

In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the Crank-Nicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math. Comp., 75 (2006), pp. 511-531], while the second one is new. Moreover, in the case of isotropic meshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195 212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm.

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Type
research article
DOI
10.1137/080715135
Web of Science ID

WOS:000268859500016

Author(s)
Lozinski, Alexei  
•
Picasso, Marco  
•
Prachittham, Virabouth
Date Issued

2009

Published in
Siam Journal On Scientific Computing
Volume

31

Start page

2757

End page

2783

Subjects

anisotropic finite elements

•

a posterori error estimates

•

Crank-Nicolson

•

Finite-Element Methods

•

Elliptic-Equations

•

Tetrahedral Meshes

•

Nonlinear Problems

•

Aspect-Ratio

•

Recovery

•

Discretizations

•

Refinement

•

Efficient

•

Model

Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
ASN  
Available on Infoscience
November 30, 2010
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/59932
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