Volume 12, Issue 3
Variational Multiscale a Posteriori Error Estimation for 2nd and 4th-Order ODEs

Int. J. Numer. Anal. Mod., 12 (2015), pp. 430-454.

Published online: 2015-12

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• Abstract

In this paper, an explicit a posteriori error estimator is developed for second and fourth order ODEs solved with the Galerkin method that, remarkably, provides exact pointwise error estimates. The error estimator is derived from the variational multiscale theory, in which the subgrid scales are approximated making use of fine-scale Green's functions. This methodology can be extended to any element type and order. Second and fourth order differential equations cover a great variety of problems in mechanics. Two examples with application in elasticity have been studied: the axially loaded beam and the Euler-Bernoulli beam. Because the error estimator is explicit, it can be very easily implemented and its computational cost is very small. Apart from pointwise error estimates, we present local and global a posteriori error estimates in the $L^1$-norm, the $L^2$-norm and the $H^1$-seminorm. Finally, convergence rates of the error and the efficiencies of the estimator are analyzed.

35R35, 49J40, 60G40

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@Article{IJNAM-12-430, author = {}, title = {Variational Multiscale a Posteriori Error Estimation for 2nd and 4th-Order ODEs}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {3}, pages = {430--454}, abstract = {

In this paper, an explicit a posteriori error estimator is developed for second and fourth order ODEs solved with the Galerkin method that, remarkably, provides exact pointwise error estimates. The error estimator is derived from the variational multiscale theory, in which the subgrid scales are approximated making use of fine-scale Green's functions. This methodology can be extended to any element type and order. Second and fourth order differential equations cover a great variety of problems in mechanics. Two examples with application in elasticity have been studied: the axially loaded beam and the Euler-Bernoulli beam. Because the error estimator is explicit, it can be very easily implemented and its computational cost is very small. Apart from pointwise error estimates, we present local and global a posteriori error estimates in the $L^1$-norm, the $L^2$-norm and the $H^1$-seminorm. Finally, convergence rates of the error and the efficiencies of the estimator are analyzed.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/497.html} }
TY - JOUR T1 - Variational Multiscale a Posteriori Error Estimation for 2nd and 4th-Order ODEs JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 430 EP - 454 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/497.html KW - a posteriori error estimation, 1D linear elasticity, Euler-Bernoulli beam, pointwise error, variational multiscale theory. AB -

In this paper, an explicit a posteriori error estimator is developed for second and fourth order ODEs solved with the Galerkin method that, remarkably, provides exact pointwise error estimates. The error estimator is derived from the variational multiscale theory, in which the subgrid scales are approximated making use of fine-scale Green's functions. This methodology can be extended to any element type and order. Second and fourth order differential equations cover a great variety of problems in mechanics. Two examples with application in elasticity have been studied: the axially loaded beam and the Euler-Bernoulli beam. Because the error estimator is explicit, it can be very easily implemented and its computational cost is very small. Apart from pointwise error estimates, we present local and global a posteriori error estimates in the $L^1$-norm, the $L^2$-norm and the $H^1$-seminorm. Finally, convergence rates of the error and the efficiencies of the estimator are analyzed.

Diego Irisarri & Guillermo Hauke. (1970). Variational Multiscale a Posteriori Error Estimation for 2nd and 4th-Order ODEs. International Journal of Numerical Analysis and Modeling. 12 (3). 430-454. doi:
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