On the accuracy of the finite element method plus time relaxation

Connors, J.; Layton, W.

doi:10.1090/S0025-5718-09-02316-3

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 , 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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