We develop a Petrov-Galerkin stabilization method for multiscale convection-diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion, which may not be sufficient to stabilize multiscale systems. We seek a local reduced-order model for this kind of multiscale transport problems and thus, develop a systematic approach for finding reduced-order approximations of the solution. We start from a Petrov-Galerkin framework using optimal weighting functions. We introduce an auxiliary variable to a mixed formulation of the problem. The auxiliary variable stands for the optimal weighting function. The problem reduces to finding a test space (a dimensionally reduced space for this auxiliary variable), which guarantees that the error in the primal variable (representing the solution) is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution. To find the test space, we reformulate some recent mixed Generalized Multiscale Finite Element Methods. We introduce snapshots and local spectral problems that appropriately define local weight and trial spaces. In particular, we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable. The resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system. We discuss the stability and its relation to the approximation property of the test space. We design online basis functions, which accelerate convergence in the test space, and consequently, improve stability. We present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the Peclet number.
|Original language||English (US)|
|Number of pages||19|
|Journal||Computer Methods in Applied Mechanics and Engineering|
|State||Published - Feb 23 2016|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work is part of the European Union’s Horizon 2020 research and innovation programme of the Marie Sklodowska-Curie grant agreement No 644602. 2014-0191. This publication also was made possible by a National Priorities Research Program grant NPRP grant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors. The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 14301314).
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Mechanics of Materials
- Mechanical Engineering
- Computational Mechanics
- Computer Science Applications