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Mathematics of Computation

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On the accuracy of the finite element method plus time relaxation


Authors: J. Connors and W. Layton
Journal: Math. Comp. 79 (2010), 619-648
MSC (2010): Primary 65M15; Secondary 65M60
DOI: https://doi.org/10.1090/S0025-5718-09-02316-3
Published electronically: December 16, 2009
MathSciNet review: 2600537
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Abstract: If $ \overline{u}$ denotes a local, spatial average of $ u$, then $ u^{\prime }=u-\overline{u}$ is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation $ u_{t}+\overrightarrow{a}\cdot\nabla u=f(x,t)$ is

$\displaystyle (u_{h,t}+\overrightarrow{a}\cdot\nabla u_{h},v_{h})+\chi(u_{h}^{\prime} ,v_{h}^{\prime})=(f(x,t),v_{h}). $

We analyze the error in this and (more importantly) higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation.


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Additional Information

J. Connors
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

W. Layton
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: wjl@pitt.edu

DOI: https://doi.org/10.1090/S0025-5718-09-02316-3
Keywords: Time relaxation, deconvolution, hyperbolic equation, finite element method
Received by editor(s): May 30, 2008
Received by editor(s) in revised form: December 13, 2008
Published electronically: December 16, 2009
Additional Notes: The work of both authors was partially supported by NSF grants DMS 0508260 and 0810385.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.