Multiscale stabilization for convection-dominated diffusion in heterogeneous media

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

12 Scopus Citations
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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)359-377
Journal / PublicationComputer Methods in Applied Mechanics and Engineering
Volume304
Publication statusPublished - 1 Jun 2016
Externally publishedYes

Abstract

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.

Research Area(s)

  • Convection-dominated diffusion, Discontinuous Petrov-Galerkin method, Generalized multiscale finite element method, Optimal weighting functions, Snapshot spaces construction

Bibliographic Note

Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

Citation Format(s)

Multiscale stabilization for convection-dominated diffusion in heterogeneous media. / Calo, Victor M.; Chung, Eric T.; Efendiev, Yalchin et al.
In: Computer Methods in Applied Mechanics and Engineering, Vol. 304, 01.06.2016, p. 359-377.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review