Volume 51, Number 2, March-April 2017
|Page(s)||641 - 677|
|Published online||24 February 2017|
Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques
Received: 29 December 2013
Revised: 16 December 2015
Accepted: 20 April 2016
In this paper we extend the hierarchical model reduction framework based on reduced basis techniques recently introduced in [M. Ohlberger and K. Smetana, SIAM J. Sci. Comput. 36 (2014) A714–A736] for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of the adaptive empirical projection method, which is an adaptive integration algorithm based on the (generalized) empirical interpolation method [M. Barrault, et al., C. R. Math. Acad. Sci. Paris Series I 339 (2004) 667–672; Y. Maday and O. Mula, A generalized empirical interpolation method: Application of reduced basis techniques to data assimilation. In Analysis and Numerics of Partial Differential Equations. Vol. 4 of Springer INdAM Series. Springer Milan (2013) 221–235]. Different from other partitioning concepts for the empirical interpolation method we perform an adaptive decomposition of the spatial domain. We project both the variational formulation and the range of the nonlinear operator onto reduced spaces. Those reduced spaces combine the full dimensional (finite element) space in an identified dominant spatial direction and a reduction space or collateral basis space spanned by modal orthonormal basis functions in the transverse direction. Both the reduction and the collateral basis space are constructed in a highly nonlinear fashion by introducing a parametrized problem in the transverse direction and associated parametrized operator evaluations, and by applying reduced basis methods to select the bases from the corresponding snapshots. Rigorous a priori and a posteriori error estimators, which do not require additional regularity of the nonlinear operator are proven for the adaptive empirical projection method and then used to derive a rigorous a posteriori error estimator for the resulting hierarchical model reduction approach. Numerical experiments for an elliptic nonlinear diffusion equation demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full-dimensional problem. Runtime experiments verify a close to linear scaling of the reduction method in the number of degrees of freedom used for the computations in the dominant direction.
Mathematics Subject Classification: 65N15 / 65N30 / 65Y20 / 35J60 / 65D05 / 65D30
Key words: Dimensional reduction / hierarchical model reduction / reduced basis methods / a posteriori error estimation / nonlinear partial differential equations / empirical interpolation / finite elements
© EDP Sciences, SMAI 2017
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.