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Volume 11, Issue 2
Recent Advances on Explicit Variational Multiscale a Posteriori Error Estimation for Systems

G. Hauke, D. Irisarri & F. Lizarraga

Int. J. Numer. Anal. Mod., 11 (2014), pp. 372-384.

Published online: 2014-11

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  • Abstract

In 1995 the genesis of stabilized methods was established by Professor Hughes from the standpoint of the variational multiscale theory (VMS). By splitting the solution into resolved and unresolved scales, it was unveiled that stabilized methods take into account an approximation of the unresolved scales or error into the finite element solution. In this work, the VMS theory is exploited to formulate an explicit a-posteriori error estimator, consistent with the assumptions inherent to stabilized methods.The proposed technology, which is especially suited for fluid flow problems, is very economical and can be implemented in standard finite element codes. It has been shown that, in practice, the method is robust uniformly from the diffusive to the hyperbolic limit.The success of the method can be explained by the fact that in stabilized methods the element local problems for the fine-scale Green's function capture most of the error and the error intrinsic time-scales are an approximation to the solution of the dual problem. Applications to the Euler and linear elasticity equations are shown.

  • AMS Subject Headings

35Q31

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-372, author = {}, title = {Recent Advances on Explicit Variational Multiscale a Posteriori Error Estimation for Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {2}, pages = {372--384}, abstract = {

In 1995 the genesis of stabilized methods was established by Professor Hughes from the standpoint of the variational multiscale theory (VMS). By splitting the solution into resolved and unresolved scales, it was unveiled that stabilized methods take into account an approximation of the unresolved scales or error into the finite element solution. In this work, the VMS theory is exploited to formulate an explicit a-posteriori error estimator, consistent with the assumptions inherent to stabilized methods.The proposed technology, which is especially suited for fluid flow problems, is very economical and can be implemented in standard finite element codes. It has been shown that, in practice, the method is robust uniformly from the diffusive to the hyperbolic limit.The success of the method can be explained by the fact that in stabilized methods the element local problems for the fine-scale Green's function capture most of the error and the error intrinsic time-scales are an approximation to the solution of the dual problem. Applications to the Euler and linear elasticity equations are shown.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/532.html} }
TY - JOUR T1 - Recent Advances on Explicit Variational Multiscale a Posteriori Error Estimation for Systems JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 372 EP - 384 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/532.html KW - A posteriori error estimation, stabilized methods. AB -

In 1995 the genesis of stabilized methods was established by Professor Hughes from the standpoint of the variational multiscale theory (VMS). By splitting the solution into resolved and unresolved scales, it was unveiled that stabilized methods take into account an approximation of the unresolved scales or error into the finite element solution. In this work, the VMS theory is exploited to formulate an explicit a-posteriori error estimator, consistent with the assumptions inherent to stabilized methods.The proposed technology, which is especially suited for fluid flow problems, is very economical and can be implemented in standard finite element codes. It has been shown that, in practice, the method is robust uniformly from the diffusive to the hyperbolic limit.The success of the method can be explained by the fact that in stabilized methods the element local problems for the fine-scale Green's function capture most of the error and the error intrinsic time-scales are an approximation to the solution of the dual problem. Applications to the Euler and linear elasticity equations are shown.

G. Hauke, D. Irisarri & F. Lizarraga. (1970). Recent Advances on Explicit Variational Multiscale a Posteriori Error Estimation for Systems. International Journal of Numerical Analysis and Modeling. 11 (2). 372-384. doi:
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