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

Diego Irisarri & Guillermo Hauke

DOI:

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 L¹-norm, the L²-norm and the H¹-seminorm. Finally, convergence rates of the error and the efficiencies of the estimator are analyzed.

  • Keywords

a posteriori error estimation 1D linear elasticity Euler-Bernoulli beam pointwise error variational multiscale theory

  • AMS Subject Headings

35R35 49J40 60G40

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-12-430, author = {Diego Irisarri and Guillermo Hauke}, 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 L¹-norm, the L²-norm and the H¹-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 AU - Diego Irisarri & Guillermo Hauke JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 430 EP - 454 PY - 2015 DA - 2015/12 SN - 12 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/ijnam/497.html KW - a posteriori error estimation KW - 1D linear elasticity KW - Euler-Bernoulli beam KW - pointwise error KW - 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 L¹-norm, the L²-norm and the H¹-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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