The partition of unity finite element method (PUFEM) proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner. The lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures. In contrast to traditional hp-adaptivity for finite elements, the proposed approach preserves discrete conservation properties and the continuity of traces at common boundaries of adjacent mesh cells. In the context of space discretizations, a continuous blending function is used to combine finite element bases corresponding to high-order polynomials and piecewise-linear approximations based on the same set of nodes. In a similar vein, spatially partitioned time discretizations can be designed using weights that depend continuously on the space variable. The design of blending functions may be based on a priori knowledge (e.g., in applications to problems with singularities or boundary layers), local error estimates, smoothness indicators, and/or discrete maximum principles. In adaptive methods, changes of the finite element approximation exhibit continuous dependence on the data. The presented numerical examples illustrate the typical behavior of local H1 and L2 errors.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The work of Dmitri Kuzmin was supported by the German Research Association (DFG), Germany under grant KU 1530/23-1. He would like to thank Hennes Hajduk (TU Dortmund University) and Friedhelm Schieweck (Otto von Guericke University Magdeburg) for helpful discussions. The work of Manuel Quezada de Luna was supported in part by an appointment to the Postgraduate Research Participation Program at the U.S. Army Engineer Research and Development Center, Costal and Hydraulics Laboratory (ERDC-CHL), USA administrated by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and ERDC. Permission was granted by the Chief of Engineers, US Army Corps of Engineers, to publish this information.