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
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by J. Connors and W. Layton PDF

Math. Comp. 79 (2010), 619-648 Request permission

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