Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods TeX Export

@article{citeulike:1288670, abstract = {We derive an explicit formula for the fine-scale Green's function arising in variational multiscale analysis. The formula is expressed in terms of the classical Green's function and a projector which defines the decomposition of the solution into coarse and fine scales. The theory is presented in an abstract operator format and subsequently specialized for the advection-diffusion equation. It is shown that different projectors lead to fine-scale Green's functions with very different properties. For example, in the advection-dominated case, the projector induced by the \$H^1\_0\$-seminorm produces a fine-scale Green's function which is highly attenuated and localized. These are very desirable properties in a multiscale method and ones that are not shared by the \$L^2\$-projector. By design, the coarse-scale solution attains optimality in the norm associated with the projector. This property, combined with a localized fine-scale Green's function, indicates the possibility of effective methods with local character for dominantly hyperbolic problems. The constructs lead to a new class of stabilized methods, and the relationship between \$H^1\_0\$-optimality and the streamline-upwind Petrov-Galerkin (SUPG) method is described.}, author = {Hughes, T. J. R. and Sangalli, G.}, citeulike-article-id = {1288670}, citeulike-linkout-0 = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000045000002000539000001&idtype=cvips&gifs=yes}, citeulike-linkout-1 = {http://link.aip.org/link/?SNA/45/539}, journal = {SIAM Journal on Numerical Analysis}, keywords = {computational-homogenization}, number = {2}, pages = {539--557}, posted-at = {2007-05-10 17:41:34}, priority = {2}, publisher = {SIAM}, title = {Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods}, url = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000045000002000539000001&idtype=cvips&gifs=yes}, volume = {45}, year = {2007} }

TY - JOUR ID - citeulike:1288670 L3 - citeulike-article-id:1288670 N2 - We derive an explicit formula for the fine-scale Green's function arising in variational multiscale analysis. The formula is expressed in terms of the classical Green's function and a projector which defines the decomposition of the solution into coarse and fine scales. The theory is presented in an abstract operator format and subsequently specialized for the advection-diffusion equation. It is shown that different projectors lead to fine-scale Green's functions with very different properties. For example, in the advection-dominated case, the projector induced by the $H^1_0$-seminorm produces a fine-scale Green's function which is highly attenuated and localized. These are very desirable properties in a multiscale method and ones that are not shared by the $L^2$-projector. By design, the coarse-scale solution attains optimality in the norm associated with the projector. This property, combined with a localized fine-scale Green's function, indicates the possibility of effective methods with local character for dominantly hyperbolic problems. The constructs lead to a new class of stabilized methods, and the relationship between $H^1_0$-optimality and the streamline-upwind Petrov-Galerkin (SUPG) method is described. IS - 2 JF - SIAM Journal on Numerical Analysis EP - 557 TI - Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods PB - SIAM VL - 45 SP - 539 KW - computational-homogenization AU - Hughes, TJR AU - Sangalli, G PY - 2007/// UR - http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000045000002000539000001&idtype=cvips&gifs=yes ER -