A partition of unity approach to adaptivity and limiting in continuous finite element methods

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.