| Speaker: | Guillermo Hauke, Universidad de Zaragoza, Spain |
| Title: | Explicit A-Posteriori Error Esitimation of Advection-Dominated Flows: The Variational Multiscale Paradigm |
| Time: | 3:00-4:00 pm |
| Place: | Blocker 628 |
Explicit a posteriori error estimation is investigated under the light of the variational multiscale theory. The methodology brings up naturally all the fundamental kinds of residuals, namely, interior residuals, inter-element jumps and the natural boundary condition residual. As a starting point, the solution error is neglected along element edges, yielding the smooth version of the multiscale error estimator. In this case, the element Green's functions are able to capture the fine scale problem. Indeed, this is an exact paradigm for element-edge exact solutions, such as one-dimensional stabilized convection-diffusion solutions or one-dimensional Galerkin Laplace solutions. It is also a good approximation for solutions where the jump contributions to the error are neglegible, such as for multi-dimensional advection-dominated flows. In first place, the methodology is applied to piecewise constant residuals and linear finite element spaces. Theoretical studies reveal the proper scales (and constants) to predict the error in various norms, such as $L_2$, $H1$, $L_\infty$. The estimates above can be combined to output error estimates in other norms, such as the energy norm. For fluid dynamics problems, it will be shown that the proper constant for error estimation in the $H1$ seminorm is not well behaved, since it tends to infinity for large element Peclet numbers. This is not the case for error estimation in the $L_p$ norms, where the constant is bounded, indicating that, for fluid dynamics, $L_p$ norms are more suitable for a posteriori error estimation than $H1$ seminorms. In second place, the technique is applied to general residuals and higher-order elements, given rise to error upper bounds,. In particular, it will be shown that the flow intrinsic time scale parameter $\tau$ estimates the solution error in the $L_1$ norm as a function of the residual in the $L_\infty$ norm. Finally, the methodology is implemented for fluid flow transport. Numerical examples confirm that the method is exact for stabilized one-dimensional linear problems and that, for multi-dimensional convection dominated flows, the efficiency index is practically independent of the diffusion coefficient, yielding to a robust a posteriori error estimator.
Last revised: 04/21/06 By: sgkim@math.tamu.edu