This is the peer reviewed version of the following article: Ibáñez, R. [et al.]. Multiscale proper generalized decomposition based on the partition of unity. "International journal for numerical methods in engineering", 9 November 2019, vol. 120, núm. 6, p. 727-747, which has been published in final form at DOI: 10.1002/nme.6154. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

Solutions of partial differential equations could exhibit a multiscale behavior. Standard discretization techniques are constraints to mesh up to the finest scale to predict accurately the response of the system. The proposed methodology is based on the standard proper generalized decomposition rationale; thus, the PDE is transformed into a nonlinear system that iterates between microscale and macroscale states, where the time coordinate could be viewed as a 2D time, representing the microtime and macrotime scales. The macroscale effects are taken into account because of an FEM-based macrodiscretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the domain. The proposed methodology can be seen as an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors.

Peer Reviewed